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resultant

resultant

resultant Sentence Examples

  • There is no difficulty in expressing the resultant by the method of symmetric functions.

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  • temperance and the resultant health and vigour).

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  • The main branches of the resultant " tree " may be rendered as follows: [[Coraciomorphae Odontolcae..Colymbo-+Pelargoalectoromorphae..Ratitae Morphae Morphae ' 'Neornithes]] The Odontolcae seem to be an early specialized offshoot of the Colymbo-Pelargomorphous brigade, while the Ratitae represent a number of side branches of early Alectoromorphae.

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  • Through the resultant scarcity of labor, much land fell out of cultivation.

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  • The general character of the country, resultant on these conditions, varies according to elevation and latitude.

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  • It is also the custom to balance a proportion of the reciprocating masses by balance weights placed between the spokes of the wheels, and the actual balance weight seen in a driving-wheel is the resultant of the separate weights required for the balancing of the revolving parts and the reciprocating parts.

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  • Another of Roberval's discoveries was a very general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.

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  • The radiations interfere in an optical sense of the word, and in some directions reinforce each other and in other directions neutralize each other, so making the resultant radiation greater in some directions than others.

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  • Hence, finally, the resultant is expressed in terms of the coefficients of the three equations, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees np and pm in the coefficients of the first and second equations respectively.

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  • principal axes of the crystal, the actual magnetization will be the resultant of the three magnetizations along the axes.

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  • To find component forces equal to the composite or resultant force, the sum of the components must equal the resultant.

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  • The historian evidently decomposes Alexander's power into the components: Talleyrand, Chateaubriand, and the rest--but the sum of the components, that is, the interactions of Chateaubriand, Talleyrand, Madame de Stael, and the others, evidently does not equal the resultant, namely the phenomenon of millions of Frenchmen submitting to the Bourbons.

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  • R is a function of the coefficients which is called the " resultant " or " eliminant " of the k equations, and the process by which it is obtained is termed " elimination."

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  • and by elimination we obtain the resultant ao 0 bo 0 0 al ao b1 bo 0 a 2 a i b 2 b 1 bo a numerical factor being disregarded.

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  • Bezout's method gives the resultant in the form of a determinant of order m or n, according as m is n.

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  • He first divides by the factor x -x', reducing it to the degree m - I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers xo, x, - of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m.

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  • = 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, corn bined with the chosen values of y and z, yields a system of values which satisfies both equations.

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  • Hence this product is the required resultant of the three equations.

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  • The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner.

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  • It is the resultant of k polynomials each of degree m-I, and thus contains the coefficients of each form to the degree (m-I)'-1; hence the total degrees in the coefficients of the k forms is, by addition, k (m - 1) k - 1; it may further be shown that the weight of each term of the resultant is constant and equal to m(m-I) - (Salmon, l.c. p. loo).

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  • This can be seen at once because the factor in question being once repeated in both differentials, the resultant of the latter must vanish.

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  • = = o, and have in consequence a vanishing resultant.

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  • af Expression in Terms of Roots.-Since x+y y =mf, if we take cx any root x 3, y1, ofand substitute in mf we must obtain, y 1 C) zaZ1 �; hence the resultant of and f is, disregarding numerical factors, y,y2...y,,.

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  • (x y m - x m y), ar _ y1(x y 2 - and substituting in the latter any root of f and forming the product, we find the resultant of f and d, viz.

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  • The discriminant of the product of two forms is equal to the product of their discriminants multiplied by the square of their resultant.

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  • Schlafli 1 this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables.

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  • This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a l, a2; b l, b 2 are alternative, are ultimately put equal to one another and (ab) vanishes.

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  • In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant.

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  • The existence of such forms seems to have been brought to Sylvester's notice by observation of the fact that the resultant of of and b must be a factor of the resultant of Xax+ 12 by and X'a +tA2 for a common factor of the first pair must be also a common factor so we obtain P: = of the second pair; so that the condition for the existence of such common factor must be the same in the two cases.

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  • Remark.-The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C = o, p is the common factor of i and j.

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  • The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.

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  • 1 A2B' Where The Denominator Factors Indicate The Forms Themselves, Their Jacobian, The Invariant Of The Quadratic And Their Resultant; Connected, As Shown By The Numerator, By A Syzygy Of Degreesorder (2, 2; 2).

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  • The process of magnetization consists in turning round the molecules by the application of magnetic force, so that their north poles may all point more or less approximately in the direction of the force; thus the body as a whole becomes a magnet which is merely the resultant of an immense number of molecular magnets.

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  • A magnet attached to a cork and [[[Terminology And Principles]] floated upon water will set itself with its axis in the magnetic meridian, but it will be drawn neither northward nor southward; the forces acting upon the two poles have therefore no horizontal resultant.

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  • terminate outside the magnet or inside, have a resultant, equal to the sum of the forces and parallel to their direction, acting at a certain point N.

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  • Similarly, the forces acting in the opposite direction on the negative poles of the filaments have a resultant at another point S, which is called the south or negative pole.

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  • The line through the given point along which the potential decreases most rapidly is the direction of the resultant magnetic force, and the rate of decrease of the potential in any direction is equal to the component of the force in that direction.

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  • If V denote the potential, F the resultant force, X, Y, Z, its components parallel to the co-ordinate axes and n the line along which the force is directed, then - sn = F, b?= X, - Sy = Y, -s Surfaces for which the potential is constant are called equipotential surfaces.

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  • For the resultant force at P, F=-VF r 2.

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  • Since 7ra'I is the moment of the sphere (=volume X magnetization), it appears from (10) that the magnetized sphere produces the same external effect as a very small magnet of equal moment placed at its centre and magnetized in the same direction; the resultant force therefore is the same as in (14).

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  • The resultant magnetic field, therefore, is compounded of two fields, the one being due to the poles, and the other to the external causes which would be operative in the absence of the magnetized metal.

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  • The intensity (at any point) of the field due to the magnetization may be denoted by H i, that of the external field by Ho, and that of the resultant field by H.

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  • Magnetization is usually regarded as the direct effect of the resultant magnetic force, which is therefore often termed the magnetizing force.

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  • Demagnetizing Force.-It has already been mentioned that when a ferromagnetic body is placed in a magnetic field, the resultant magnetic force H, at a point within the body, is compounded of the force H o, due to the external field, and of another force, Hi, arising from the induced magnetization of the body.

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  • Equations (33) and (34) show that when, as is generally the case with ferromagnetic substances, the value of is considerable, the resultant magnetic force is only a small fraction of the external force, while the numerical value of the induction is approximately three times that of the external force, and nearly independent of the permeability.

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  • The action of a hollow magnetized shell on a point inside it is always opposed to that of the external magnetizing force, 6 the resultant interior field being therefore weaker than the field outside.

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  • 5), we have for the resultant force at P R = - 2cosO X m/d' = - elm/d 3 = - M/d3.

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  • The direction of the resultant field of force will then make, with that of H E, an angle 0, such that Hp/H E tan 0, and the suspended needle will be deflected through the same angle.

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  • Then, since Hp, the force at M due to m and - m, is the resultant of d, and - m2, we have (L Hp m d d2 2ml H = d3, the direction being parallel to AB.

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  • The wire is subject to two superposed magnetizations, the one longitudinal, the other circular, due to the current traversing the wire; the resultant magnetization is consequently in the direction of a screw or spiral round the wire, which will be right-handed or left-handed according as the relation between the two magnetizations is right-handed or left-handed; the magnetic expansion or contraction of the metal along the spiral lines of magnetization produces the Wiedemann twist.

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  • Weber therefore supposed each molecule to be acted on by a force tending to preserve it in its original direction, the position actually assumed by the axis being in the direction of the resultant of this hypothetical force and the applied magnetizing force.

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  • many different directions, so that there is no resultant magnetic moment.

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  • If now a gradually increasing magnetizing force is applied, the needles at first undergo a stable deflection, giving to the group a small resultant moment which increases uniformly with the force; and if the current is interrupted while the force is still weak, the needles merely return to their initial positions.

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  • The rearrangement is completed within a comparatively small range of magnetizing force, a rapid increase of the resultant moment being thus brought about.

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  • When the field is removed, many of the newly formed combinations are but slightly disturbed, and the group may consequently retain a considerable resultant moment.

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  • This corresponds to the second stage of magnetization, in which the susceptibility is large and permanent magnetization is set up. A still stronger magnetizing force has little effect except in causing the direction of the needles to approach still more nearly to that of the field; if the force were infinite, every member of the group ‘ would have exactly the same direction and the greatest possible resultant moment would be reached; this illustrates " magnetic saturation " - the condition approached in the third stage of magnetization.

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  • If the structure of the molecule is so perfectly symmetrical that, in the absence of any external field, the resultant magnetic moment of the circulating electrons is zero, then the application of a field, by accelerating the right-handed (negative) revolutions, and retarding those which are left-handed, will induce in the substance a resultant magnetization opposite in direction to the field itself; a body composed of such symmetrical molecules is therefore diamagnetic. If however the structure of the molecule is such that the electrons revolving around its atoms do not exactly cancel one another's effects, the molecule constitutes a little magnet, which under the influence of an external field will tend to set itself with its axis parallel to the field.

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  • The phase of the resultant is midway between those of the extreme elements, that is to say, a quarter of a period behind that due to the element at the centre of the circle.

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  • The middle element alone contributes without deduction; the effect of every other must be found by introduction of a resolving factor, equal to cos 0, if 0 represent the difference of phase between this element and the resultant.

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  • Accordingly, the amplitude of the resultant will be less than if all its components had the same phase, in the ratio +17r -17r or 2: 7.

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  • The process of augmenting the resultant illumination at a particular point by stopping some of the secondary rays may be carried much further (Soret, Pogg.

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  • When the difference of phase amounts to A, we may expect the resultant illumination to be very much reduced.

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  • The phase of the resultant amplitude is the same as that due to the central secondary wave, and the discrepancies of phase among the components reduce the amplitude in the proportion l ` dri): 3?

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  • Since at these points the resultant due to the whole aperture is zero, any two portions into which the whole may be divided must give equal and opposite resultants.

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  • The effect of each of the elements of the grating is then the same; and, unless this vanishes on account of a particular adjustment of the ratio a: d, the resultant amplitude becomes comparatively very great.

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  • The phase of the resultant effect is by symmetry that of the component which comes from the middle of a.

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  • The vibrations corresponding to the two parts are precisely antagonistic, since if both were operative the resultant would be zero.

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  • If the resultant rotation be n, we have TZ iJ (x 2 -{-y 2) de ikr TZsin4 d e ikr 2 r ' dr (r !

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  • For if the body is removed, and replaced by the fluid as at first, this fluid is in equilibrium under its own weight and the thrust of the surrounding fluid, which must be equal and opposite, and the surrounding fluid acts in the same manner when the body replaces the displaced fluid again; so that the resultant thrust of the fluid acts vertically upward through the centre of gravity of the fluid displaced, and is equal to the weight.

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  • The resultant vertical thrust on any portion of a curved surface exposed to the pressure of a fluid at rest under gravity is the weight of fluid cut out by vertical lines drawn round the boundary of the curved surface.

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  • The resultant horizontal thrust in any direction is obtained by drawing parallel horizontal lines round the boundary, and intersecting a plane perpendicular to their direction in a plane curve; and then investigating the thrust on this plane area, which will be the same as on the curved surface.

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  • The resultant force is therefore in the direction of the steepest pressure-gradient, and this is normal to the surface of equal pressure; for equilibrium to exist in a fluid the lines of force must therefore be capable of being cut orthogonally by a system of surfaces, which will be surfaces of equal pressure.

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  • - A plane area exposed to fluid pressure on one Side experiences a single resultant thrust, the integrated pressure over the area, acting through a definite point called the centre of pressure (C.P.) of the area.

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  • Thus if the plane is normal to Or, the resultant thrust R =f fpdxdy, (r) and the co-ordinates x, y of the C.P. are given by xR = f f xpdxdy, yR = f f ypdxdy.

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  • tan 0, so that the resultant righting couple is W.GM.

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  • Putting (12) a vortex line is defined to be such that the tangent is in the direction of w, the resultant of, n, called the components of molecular rotation.

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  • If other vortices are present, any one may be supposed to move with the velocity due to the others, the resultant stream function being = gy m log r =log IIrm; (9) the path of a vortex is obtained by equating the value of 1P at the vortex to a constant, omitting the rm of the vortex itself.

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  • When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (U 2 + V2)b2a a2U+Vi +b a b a2 U z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (U b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?

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  • Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Ua 2 r2 cos 20 +mr i sin 0) (Ua 2 r 2 sin 2 0+mr 1 cos 0) + pdy (- U+Ua 2 r 2 cos 2 0 +mr1 sin 0) (Ua 2 r 2 sin 2 0 -mr 1 cos 0) =2pdymUr '(cos 0 -a 2 r 2 cos 30), (8) and with b tan r =b sec this is 2pmUdo(i -a 2 b2 cos 30 cos 0), (9) and integrating between the limits 0 = 27r, the resultant, as before, is 27rpmU.

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  • having a resultant in the direction PO, where P is the intersection of an ellipse n with the hyperbola 13; and with this velocity the ellipse n can be swimming in the liquid, without distortion for an instant.

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  • The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin Olga, and its thrust is 27rpmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r=a with velocity U reversed, and the cylinder is surrounded by a vortex.

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  • If at any points of a vortex line the cross-section ABC, A'B'C' is drawn of the vortex filament, joined by the vortex line AA', then, since the flow in AA' is taken in opposite directions in the complete circuit ABC AA'B'C' A'A, the resultant flow in AA' cancels, and the circulation in ABC, A'B'C' is the same; this is expressed by saying that at all points of a vortex filament wa is constant where a is the cross-section of the filament and w the resultant spin (W.

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  • The impulse required to set up the motion in liquid of density p i the resultant of an impulsive pressure p4) over the surface S of th ellipsoid, and is therefore ffp4ldS = p4GoffxldS =p 40 (volume of the ellipsoid) =4)oW', (23) where W' denotes the weight of liquid displaced.

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  • These equations are proved by taking a line fixed in space, whose direction cosines are 1, then dt=mR-nQ,' d'-t = nP =lQ-mP. (5) If P denotes the resultant linear impulse or momentum in this direction P =lxl+mx2+nx3, ' dP dt xl+, d y t x2' x3 +1 dtl dt 2 +n dt3, =1 ('+m (dt2-x3P+x1R) ' +n ('-x1Q-{-x2P) ' '= IX +mY+nZ, / (7) for all values of 1, Next, taking a fixed origin and axes parallel to Ox, Oy, Oz through 0, and denoting by x, y, z the coordinates of 0, and by G the component angular momentum about 1"2 in the direction (1, G =1(yi-x2z+x3y) m 2-+xlz) n(y(y 3x 1 x3x y + x 2 x) (8) Differentiating with respect to t, and afterwards moving the fixed.

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  • are the components of a constant vector having a fixed direction; while (4) shows that the vector resultant of y, y, y moves as if subject to a couple of components x Wx V, x Ux W, x V-x U, (Io) and the resultant couple is therefore perpendicular to F, the resultant of x, x, x, so that the component along OF is constant, as expressed by (iii).

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  • In the absence of a medium the inertia of the body to transtion is the same in all directions, and is measured by the (3) But the change of the resultant momentum F of the medium as.

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  • Consider a submarine boat or airship moving freely with the direction of the resultant momentum horizontal, and the axis at a slight inclination 0.

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  • For the body alone the resultant of the components of momentum W V -cos andW V sin 0 is W V -sec. lb, acting along 00', and so is unaltered.

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  • the moment of inertia of the body about the axis, denoted by But if is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k', If the shot is spinning about its axis with angular velocity p, and is precessing steadily at a rate about a line parallel to the resultant momentum F at an angle 0, the velocity of the vector of angular momentum, as in the case of a top, is C i pµ sin 0- C2µ 2 sin 0 cos 0; (4) and equating this to the impressed couple (multiplied by g), that is, to gN = (c 1 -c 2)c2u 2 tan 0, (5) and dividing out sin 0, which equated to zero would imply perfect centring, we obtain C21 2 cos 0- (c 2 -c 1)c2u 2 sec 0 =o.

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  • The body is held fixed, and the reactio of the mechanism and the resultant of the impulsive pressure on th surface are a measure of the impulse, linear,, , and angula A, µ, v, required to start the circulation.

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  • The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by the resultant axial linear momentum of the circulation.

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  • If a small conducting body is charged with Q electrostatic units of electricity, and placed in any electric field at a point where the electric force has a value E, it will be subject to a mechanical force equal to QE dynes, tending to move it in the direction of the resultant electric force.

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  • We may otherwise define it by saying that a line of electric force is a line so drawn in a field of electric force that its direction coincides at every point with the resultant electric force at that point.

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  • It can be shown that the resultant electric force normal to the surface at a point just outside a conductor is 1 See Maxwell, Elementary Treatise on Electricity (Oxford, 1881), P. 47.

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  • Hence we see that if the whole surface of the sphere is divided into pairs of elements by cones described through any interior point, the resultant force at that point must consist of the sum of pairs of equal and opposite forces, and is therefore zero.

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  • Since then they are all charged with the same quantity of electricity, and the total over all potential difference V is the sum of each of the individual potential differences V1, V2, V3, &c., we have Q=C I V I =C 2 V 2 =C 3 V 3 =&c., and V=V1-FV2+V3+&c. The resultant capacity is C = Q/V, and C= I/(I/C1 +I /C2+1/C3+&c) = I/Z(I /C) (15).

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  • These rules provide means for calculating the resultant capacity when any number of condensers are joined up in any way.

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  • The resultant electric force E at that point is then obtained by differentiating V, since E = - dV/dx, and E is in the direction in which V diminishes fastest.

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  • In any case, therefore, in which we can sum up the elementary potentials at any point we can calculate the resultant electric force at the same point.

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  • At each point in the field the electric force can have but one resultant value.

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  • At each point on this surface the resultant force has a certain value, and a certain direction inclined at an angle 0 to the normal to the selected surface at that point.

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  • Let E be the resultant electric force at any point in the field.

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  • Then the potential at any point P in this ideal plane PO is equal to q/AP-q/BP=0, whilst the resultant force at P due to the two point charges is 2gAO/AP 3, and is parallel to AB or normal to PO.

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  • The resultant force due to these two pointcharges must then be in the direction CP, and its value E is the vector sum of the two forces along AP and BP due to the two point-charges.

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  • If, therefore, the walls of the enclosure held the gas that is directly in contact with them, this equilibrium would be the actual state of affairs; and it would follow from the principle of Archimedes that, when extraneous forces such as gravity are not considered, the gas would exert no resultant force on any body immersed in it.

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  • When a shaft is driven by means of gearing the driving torque is measured by the product of the resultant pressure P acting between the wheel teeth and the radius of the pitch circle of the wheel fixed to the shaft.

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  • (5) If c is the resultant velocity of a molecule, so that c 2 =u2+v2+w2, it is readily found from formula (4) that the number of molecules of the first kind of which the resultant velocity lies between c and c+dc is 4lrs1,l (h 3 rn 3 17r 3)e hmc2 c 2 dc. (6) These formulae express the " law of distribution of velocities " in the normal state: the law is often called Maxwell's Law of Distribution.

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  • But it can be shown that from the aggregation of these separate short motions the particle ought to have a resultant motion, described with an average velocity which, although much smaller than 2 mm.

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  • Even the resultant of mechanical forces refuses to resolve itself into its constituents.

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  • In the " resultant " there is a new direction, and with it a new quality the component forces of which no analysis can discover.'

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  • They adapted to these conditions some of the methods for managing local affairs with which they had been familiar in England, and called the resultant institution a town.

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  • HH' = 60 X 60' the resultant angle of deflection is HFH', and this can be determined by the same formula a = h X1200 X 3, but in this case h = HH' = R R 60 60 R X 3600 a _ = i' so that if the sight is inclined to the left I° it will R X 3600 give 1' deflection for every degree of elevation.

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  • the resultant with amplitude I/-/2 that of (1).

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  • Another movement of 4A in each direction gives (3) with resultant a straight line, and so on for (4) and (5).

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  • Since the resultant is zero, mU 2 sin 4) - T sin ct.

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  • If the two interfering waves, being still of same length X, be in opposite phases, or sõ that one is in advance of the other by 2X, and consequently one produces in the air the opposite state of motion to the other, then the resultant wave is one of the same length X, but the excursions of the particles are decreased, being the difference between those due to the component waves as in fig.

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  • If we measure the time from an instant at which the two are in the same phase the resultant disturbance is y=a sin i t+a sin 27rn2t =2a cos ir(n i - n 2)t sin ir(nl-t-n2)t, which may be regarded as a harmonic disturbance of frequency (ni+n2)/2 but with amplitude 2a cos 7r(n i - n 2)t slowly varying with the time.

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  • Sovereignty is a resultant of many forces.

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  • When girders form the superstructure, the resultant pressure on the piers or abutments is vertical, and the dimensions of these are simply regulated by the sufficiency to bear this vertical load.

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  • When arches form the superstructure, the abutment must be so designed as to transmit the resultant thrust to the foundation in a safe direction, and so distributed that no part may be unduly compressed.

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  • When the resultant pressure is not vertical on the piers these must be constructed to meet the inclined pressure.

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  • beyond C, the shear at C will probably be greatest when W2 is at C. Let R be the resultant of the loads on the bridge when W 1 is at C. Then the reaction at B and shear at C is Rn/l.

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  • 46) be the section at which the greatest shear is required, and let the loads advance from the left till W I is at C. If R is the resultant of the loads then on the girder, the reaction at B and shear at C is Rn/l.

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  • 70, at a distance x from the vertex, the horizontal component of the resultant (tangent to the curve) will be unaltered; the vertical component V will be simply the sum of the loads between 0 and F, or wx.

    0
    0
  • resultant tension along the chain at F, the vertical force V passing through the point D, and the horizontal tension at O; hence H: V = DC: FC = wx 2 /2 y: wx = x/2.

    0
    0
  • The nature of the resultant concepts belongs to the great controversy between Nominalism, Realism and Conceptualism.

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    0
  • The most fundamental experimental confirmation that the theory of the aether has received on the optical side in recent years has been the verification of Maxwell's proposition that radiation exerts mechanical force on a material system, on which it falls, which may be represented in all cases as the resultant of pressures operating along the rays, and of intensity equal at each point of free space to the density of radiant energy.

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  • 26-27; the length of the ministry was fixed, with some approach to certainty, at between two and three years, and here too the resultant date for the Crucifixion would be the Passover of A.D.

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  • Another question with which the textual critic of modern authors must be prepared to deal is the relative importance of different editions, each of which may have a prima facie claim to be considered authentic. Thus Shakespearean criticism must decide between the evidence of the first folio and the quartos: the critic of Shelley's poems must consider what weight is to be attached to the readings in the posthumous edition by Mrs Shelley, and in unpublished transcripts of various poems. Where there is great or complicated divergence between the editions, as in the case of Marlowe's Faustus, the production of a resultant text which may be relied upon to represent the ultimate intention of the author is well-nigh impossible.

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  • The umbrella analogy is similarly explained; the most efficient position being when the stick points along the resultant AD.

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    0
  • A full description of its modern condition is therefore given under the heading Oviedo; the present article being confined to an account of its physical features, its history, and the resultant character of its inhabitants.

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    0
  • What is done is to divide the resultant force due to gravitation into two components, one of which corresponds to this acceleration, while the other one is what is called the "weight" of the body.

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    0
  • The distinction between these two classes is not sharp; though when the properties of the resultant are sensibly the sum of those of the pure components, as is nearly the case for a complex gas such as air, it is usual to class it as a mixture.

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    0
  • When the properties of the resultant substance are different from those of the components and it is not a chemical compound we define it as a solution.

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    0
  • By cooling the resultant solution through the range dT we recover the original state of the system.

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  • In 1856 Bessemer not only invented his extraordinary process of making the heat developed by the rapid oxidation of the impurities in pig iron raise the temperature above the exalted melting-point of the resultant purified steel, but also made it widely known that this steel was a very valuable substance.

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  • Slag or Cinder, a characteristic component of wrought iron, which usually contains from 0.20 to 2.00% of it, is essentially a silicate of iron (ferrous silicate), and is present in wrought iron simply because this product is made by welding together pasty granules of iron in a molten bath of such slag, without ever melting the resultant mass or otherwise giving the envelopes of slag thus imprisoned a chance to escape completely.

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  • Part of the resultant carbonic acid is again deoxidized to carbonic oxide by the surrounding fuel, CO 2 + C = 2C03 and the carbonic oxide thus formed deoxidizes more iron oxide, &c. As indicated in fig.

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  • Moreover, the quality of the resultant steel depends upon the temperature of the process, and this in turn depends upon the proportion of silicon, the combustion of which is the chief source of the heat developed.

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  • In the basic Bessemer process, also, unforeseen variations in the siliconcontent are harmful, because the quantity of lime added should be just that needed to neutralize the resultant silica and the phosphoric acid and no more.

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  • so powerful an acid as silica, then the phosphoric acid has so feeble a hold on the base in the slag that it is immediately redeoxidized by the carbon of the metal, or even by the iron itself, P 2 O 5 +5Fe = 2P+5FeO, and the resultant deoxidized phosphorus immediately recombines with the iron.

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  • - Silicon cannot here be used as the chief source of heat as it is in the acid Bessemer process, because most of the heat which its oxidation generates is consumed in heating the great quantities of lime needed for neutralizing the resultant silica.

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  • Further objections to the presence of silicon are that the resultant silica (1) corrodes the lining of the converter, (2) makes the slag froth so that it both throws much of the charge out and blocks up the nose of the converter, and (3) leads to rephosphorization.

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  • But Massenez and Richards, following the plan outlined by Pourcel in 1879, have found that even 3% of silicon is permissible if, by adding iron ore, the resultant silica is made into a fluid slag, and if this is removed in the early cool part of the process, when it attacks the lining of the converter but slightly.

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  • Manganese to the extent of 1.80% is desired as a means of preventing the resultant steel from being redshort, i.e.

    0
    0
  • The proportion of pig to scrap used depends chiefly on the relative cost of these two materials, but sometimes in part also on the carbon-content which the resultant steel is to have.

    0
    0
  • The ebullition from the formation of carbonic oxide puffs up the resultant phosphoric slag enough to make most of it run out of the furnace, thus both removing the phosphorus permanently from danger of being later deoxidized and returned to the steel, and partly freeing the bath of metal from the heat-insulating blanket of slag.

    0
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  • Floating on top of the molten metal, it rapidly oxidizes its phosphorus, and the resultant phosphoric acid combines with the lime in the overlying slag as phosphate of lime.

    0
    0
  • It is practically unattainable in the open-hearth furnace, because here the oxygen of the furnace atmosphere indirectly oxidizes the carbon of the metal which is kept boiling by the escape of the resultant carbonic oxide.

    0
    0
  • The resultant of these two effects has not yet been well established; but it is probable that the strongest cast iron has a little more than 1% of carbon combined as cementite, so that its matrix is nearly equivalent to the strongest of the steels.

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  • 8d.) per ton, and in very large quantities at $15 (£3, 2s.) per ton, in the latter case, according to Mr Carnegie, without further loss than that represented by interest, although the cost of each ton includes that of mining 2 tons of ore and carrying them moo miles, mining and coking 1.3 tons of coal and carrying its coke 50 m., and quarrying one-third of a ton of limestone and carrying it 140 m., besides the cost of smelting the ore, converting the resultant cast iron into steel, and rolling that steel into rails.

    0
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  • They went back to the later period of Plato's thought, the period when Plato endeavoured to combine his doctrine of Ideas with the Pythagorean number-theory, and identified the Good with the One, the source of the duality of the Infinite and the Measured (rd .bretpov and 71-pas) with the resultant scale of realities from the One down to the objects of the material world.

    0
    0
  • Sometimes the qualities common to all the gods were abstracted, and the resultant notion spoiten of as the god.

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    0
  • The brevity of the interval between the tap on the knee and the beginning of the resultant contraction of the muscle seems such as to exclude the possibility of reflex development.

    0
    0
  • The unification of the peoples of antiquity in the Roman Empire, and the resultant amalgam of religions, gave a powerful impetus to the custom.

    0
    0
  • A fluid drachm of friar's balsam may be added to a pint of water at a temperature of about 140° F., and the resultant vapour may be inhaled from the spout of a kettle or from a special inhaler.

    0
    0
  • The Streams. motion of the stars in the mean towards Canis Major is thus a resultant motion, which, when examined more minutely, is found to be due to the intermingling of two great streams of stars moving in very different directions.

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    0
  • The small minority which still retained the name joined the Original Seceders in 1842, the resultant body assuming the designation of United Original Seceders.

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    0
  • A dissentient remnant (eight congregations) of the General Associate Synod united with the Constitutional Associate Presbytery in 1827, the resultant body being called the Associate Synod of Original Seceders.

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    0
  • resultant determinate judgments, presumes of course Doivi the doctrine of the interpenetration of ideas laid down in the Sophistes as the basis of predication, but its use precedes the positive development of that formula, though not, save very vaguely, the exhibition of it, negatively, in the antinomies of the one and the many in the Parmenides.

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    0
  • Yet the resultant impression left by the whole treatment is not Herbartian.

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    0
  • Contradictions must be annulled by complementation, with resultant increasing coherence in ascending stages.

    0
    0
  • The resultant intermingling of forms is very marked and characteristic of the Pacific Coast states.

    0
    0
  • is called the resultant.

    0
    0
  • -, HK be vectors representing the given forces, the resultant will be given by AK.

    0
    0
  • If, in particular, the point K coincides with A, so that the resultant vanishes, the given system of forces is said to be FIG.

    0
    0
  • These two forces X, Y, may be combined into a single resultant R making an angle 4> with Ox, provided X = Rcos4,, Y = Rsin4~, (2) whence R2 = X~ + Yi, tan 4) = Y I X.

    0
    0
  • These can again be combined into a single resultant R acting in the direction (X, u, v), provided X=RX,YRu,ZRv.

    0
    0
  • Hence if forces P, Q act in OA, OB, the resultant R will pass through C, provided m=P/OA, n=Q/OB;

    0
    0
  • These formulae give a means of constructing the resultant by means of any transversal AB cutting the lines of action.

    0
    0
  • If we now imagine the point 0 to recede to infinity, the forces P, Q and the resultant R are parallel, and we have R=P+Q, P.AC=Q.CB.

    0
    0
  • The investigation fails when P+Q~O, since it leads to an infinitely small resultant acting in an infinitely distant line.

    0
    0
  • The sum of the moments of two forces about any point 0 is equal to the moment of their resultant (P. Varignon, 1687).

    0
    0
  • 16) represent the two forces, AD their resultant; we have to prove that the sum of the triangles OAB, OAC is D equal to the triangle OAD,

    0
    0
  • It follows that the single resultant to which the system in general reduces is uniquely determinate, i.e.

    0
    0
  • If F, Q, R, be any three forces acting along BC, CA, AB, respectively, the line of action of the resultant is determined by the consideration that the sum of the moments about any point on it must vanish.

    0
    0
  • The value of G will in general vary with the position of 0, and will vanish when 0 lies on the line of action of the single resultant.

    0
    0
  • the system reduces to a single resultant through 0, provided E.)+~.~(X)+~(xYyX)=o.

    0
    0
  • (9) If ~, ~l be regarded as current co-ordinates, this is the equation of the line of action of the single resultant to which the system is in general reducible.

    0
    0
  • so as to remain parallel, the value of 0 is alone altered, and the resultant ~(P) passes always through the point x_~(p) ~Y ~)

    0
    0
  • Hence there is a certain point, fixed relatively to the assemblage, through which the resultant of gravitational action always passes; this resultant is moreover equal to the sum of the forces on the several particles.

    0
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  • 26, where the resultant of the four given forces is denbted by R).

    0
    0
  • Now x, x have a resultant through H, represented in magnitude and direction by 00, whilst y,y have a resultant through K represented in magnitude and direction by OO.

    0
    0
  • We have only to construct the line of action of the resultant for each of two arbitrary directions of the forces; the intersection of the two lines gives the point required.

    0
    0
  • 39) are symmetrically equal, and the angle of the resultant rotation, viz.

    0
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  • The points whose displacements are in the direction of the resultant axis of rotation are determined by Ix: Iy: Iz = E: ~: ~, or (X +nz ly)IE = Cu + Ix ~z)/v = (e +:~y ~x)ft.

    0
    0
  • We assume that the body receives arbitrary twists about twc given screws, and it is required to determine the character of the resultant displacement.

    0
    0
  • The equations (7) of the axis of the resultant screw then reduce to - = y/ii, z(f2 + ~~1) = (k h)li,.

    0
    0
  • Hence, whatever the ratio f: n, the axis of the resultant screw lies on the conoidal surface z(x1+y1)=cx3~ (13)

    0
    0
  • Again, the pitch of the resultant screw is p = (X~+un)/(f2+n1) h cosio+k sin2O.

    0
    0
  • Three-dimensional Statics.A system of, parallel forces can be combined two and two until they are replaced by a single resultant equal to their sum, acting in a certain line.

    0
    0
  • In general, however, a three-dimensional system of forces cannot be replaced by a single resultant force.

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  • BB each into two compon FIG A2 ents parallel and perpendicular toOC, we see that the former components have a single resultant in OC, of amount R~AA cos a+BB cos $, (2)

    0
    0
  • Since two concurrent forces and their resultant obviously project into two concurrent forces and their resultant, we see that the sum of the moments of two concurrent forces about any axis HK is equal to the moment of their resultant.

    0
    0
  • The components of the resultant force R are unaltered, but the new components of couple are found to be L=LyZ+zY,~

    0
    0
  • By properly choosing 0 we can make the plane of the couple perpendicular to the resultant force.

    0
    0
  • Since the moment of the resultant couple is now X, Y, Z~, LX+MY+NZ

    0
    0
  • When parallel forces of given magnitudes act at given points, the resultant acts through a definite point, or centre of parallel forces, which is independent of the special direction of the forces.

    0
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  • If we imagine a rigid body to be acted on at given points by forces of given magnitudes in directions (not all parallel) which are fixed in space, then as the body is turned about the resultant wrench will assume different configurations in the body, and will in certain positions reduce to a single force.

    0
    0
  • The total work done by two concurrent forces acting on a particle, or on a rigid body, in any infinitely small displacement, is equal to the work of their resultant.

    0
    0
  • 46) represent the forces, AD their resultant, and let AH be the direction of the displacement as of the point A.

    0
    0
  • It is also evident that the total work done in two or more successive infinitely small displacements is equal to the work done in the resultant displacement.

    0
    0
  • will have a resultant represented by Z(m).OG.

    0
    0
  • Pn have a resultant proportional to ~(m) which acts always through a point G fixed relatively to the given mass-system.

    0
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  • experience the same resultant displacement PQ in the same time.

    0
    0
  • If we denote the resultant velocity at any instant by i we have j2ti+3iii~i_2gy, (7)

    0
    0
  • The motion is therefore the resultant of two simple vibrations in perpendicular directions, of periods 22r ./ (pug), 2ir ~I (pf/g).

    0
    0
  • If there are no extraneous forces, the resultant linear momentum is constant in every respect.

    0
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  • In the time ~5t a certain impulse is given to the first particle in the direction (say) from P to Q, whilst an equal and opposite impulse is given to the second in the, direction from Q to P. Since these impulses produce equal and opposite momenta in the two particles, the resultant lineal momentumof the system is unaltered.

    0
    0
  • If extraneous forces act, it is seen in like manner that the resultant linear momentum of the system is in any given time modified by the geometric additiofi of the total impulse of the extraneous forces.

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    0
  • If there are no extraneous forces, or if the extraneous forces have zero moment about any axis through G, the vector which represents the resultant angular momentum relative to G is constant in every respect.

    0
    0
  • It is to be carefully noticed that the axis of resultant angular momentum about 0 does not in general coincide with the instantaneous axis of rotation.

    0
    0
  • The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis.

    0
    0
  • Again, if F be the resultant angular momentum, so that ~ii~,ijs (10)

    0
    0
  • where I is the resultant angular momentum about 0.

    0
    0
  • Further, on examining the small variation in i/i, it appears that in a slightly disturbed slow precession the motion of any point of the axis consists of a rapid circular vibration superposed on the steady precession, so that the resultant path has a trochoidal character.

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    0
  • The former is, in fact, equal to 2T, and the latter to ~2, where T is the kinetic energy an.d r the resultant angular momentum.

    0
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  • .V,Then the pressure exerted by a structure on the earth (to which the earths resistance is equal and opposite) consists either of one pressure, which is necessarily the resultant of the weight of the structure and of all the other forces applied to it, or of two or more parallel vertical forces, whose amount can be determined at the outset of the investigation, the resistance of the earth can be treated as one or more upward loads applied to the structure.

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  • But in other cases the earth is to be treated as one of the pieces of the structure, loaded with a force equal and opposite in direction and position to the resultant of the weight of the structure and of the other pressures applied to it.

    0
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  • Re- d peating this process for each d block in succession there will be found the centres of pres sure C2, C3, &c., and also the resultant pressures R2, R3, FIG.

    0
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  • The resultant of a parallel projection of any system of forces is the projection of their resultant; and the centre of gravity of a parallel projection of a solid is the projection of the centre of gravity of the first solid.

    0
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  • Conditions of Stiffness and Strength.After the arrangement of the pieces of a structure and the size and figure of their joints or surfaces of contact have been determined so as to fulfil the conditions of stabilityconditions which depend mainly on the position and direction of the resultant or total load on each piece, and the relative magnitude of the loads on the different piecesthe dimensions of each piece singly have to be adjusted so as to fulfil the conditions of stiffness and strengthconditions which depend not only on the absolute magnitude of the load on each piece, and of the resistances by which it is balanced, but also on the mode of distribution of the load over the piece, and of the resistances over the joints.

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  • So far as the resultant velocity ratio is concerned, the order of the drivers N and of the followers n is immaterial: but to secure equable wear of the teeth, as explained in 44, the wheels ought to be so arranged that, for each elementary combination, the greatest common divisor of N and ii shall be either 1, or as small as possible.

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  • General Pr-inciples.----Willis designated as aggregate combinations those assemblages of pieces of mechanism in which the motion of one follower is the resultant of component motions impressed on it by more than one driver.

    0
    0
  • Then the motion of A relatively to the frame C is the resultant of the motion of A relatively to B and of B relativel~to C; and that resultant is to be found by the principles already explained in Division 3 of this Chapter f 2732.

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  • (I) to produce aggregate velocity, or a velocity which is the resultant of two or more components in the same path, and (2) to produce an aggregate paththat is, to make a given point in a rigid body move in an assigned path by communicating certain motions to other points in that body.

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  • The mode of distribution of a force applied to a solid body requires to be considered when its stiffness and strength are treated of; but, in questions respecting the action of a force upon a rigid body considered as a whole, the resultant of the distributed force, determined according to the principles of statics, and considered as acting in a single line and applied at a single point, may, for the occasion, be substituted for the force as really distributed.

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  • Balanced Lateral Pressure of Guides and Bearings.The most important part of the lateral pressure on a piece of mechanism is the reaction of its guides, if it is a sliding piece, or of the bearings of its axis, if it is a turning piece; and the balanced portion of this reaction is equal and opposite to the resultant of all the other forces applied to the piece, its own weight included.

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  • The total pressure exerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and its obliquity, or inclination to the common perpendicular of the surfaces, is the angle of repose formerly mentioned in 14, whose tangent is the coefficient of friction.

    0
    0
  • In oruer to diminish that pressure to the smallest possible amount, the effort, and the resultant of the useful resistance, and the weight of the piece (called above the given force) ought to be opposed to each other as directly as is practicable consistently with the purposes of the machine.

    0
    0
  • The particles of a rotating body exert centrifugal forces on each other, which strain the body, and tend to tear it asunder, but these forces balance each other, and do not affect the resultant centrifugal force exerted on the axis of rotation.i -

    0
    0
  • When the spindle 1 D is at rest the bob hangs close to it; when the spindle rotates, the bob, being made ~ c to revolve round it, diverges until the resultant of the centrifugal force and the weight of the bob is a force acting atO in the direction OB, and then it revolves steadily in a circle.

    0
    0
  • According to the principles of statics, the resultant of the force P, applied at G perpendicular to the plane OG, and the couple M is a force equal and parallel to P, but applied at a distance GC from G, in the prolongation of the perpendicular OG, whose value is GC = M/P = R2/OG.

    0
    0
  • For a given link, this force isthe resultant of all the accelerating forces distributed through the substance of the material of the link required to produce the requisite acceleration of each particle, and the determination of this force depends upon the principles of the two preceding sections.

    0
    0
  • Hence the magnitude F and the position of F relatively to the centre of gravity of the link, necessary to give rise to the couple M, are known, and this force is therefore the resultant force required.

    0
    0
  • Z is a point in the line of action of the resultant force F; hence through Z draw a line parallel to 0g.

    0
    0
  • Certain particles go forth from the eye to meet similar particles given forth from the object, and the resultant contact constitutes vision.

    0
    0
  • The resultant discontent found expression in the cry of " Portugal for the Portuguese " and in the demand for a constitution.

    0
    0
  • In the Supplement to the Theory of Capillary Action, Laplace deduced the equation of the surface of the fluid from the condition that the resultant force on a particle at the surface must be normal to the surface.

    0
    0
  • Instead of calculating the direction and magnitude of the resultant force on each particle arising from the action of neighbouring particles, he formed a single expression which is the aggregate of all the potentials arising from the mutual action between pairs of particles.

    0
    0
  • It is found by experiment that it is only very close to the bounding surface of a liquid that the forces arising from the mutual action of its parts have any resultant effect on one of its particles.

    0
    0
  • The circumference of the edge is 27rr, so that the resultant of this tension is a force 27rrT cos a acting vertically upwards on the liquid.

    0
    0
  • Hence the liquid will rise in the tube till the weight of the vertical column between the free surface and the level of the liquid in the vessel balances the resultant of the surface-tension.

    0
    0
  • Equating this force with the resultant of the tension 7rpgr 2 h = 21rrT cos a, or h = 2T cos a/pgr.

    0
    0
  • Hence the resultant of the surface-tension is 2l T cos a.

    0
    0
  • In all these cases the internal pressure exceeds the external by 2T/a where a is the semi-transverse axis of the conic. The resultant of the internal pressure and the surface-tension is equivalent to a tension along the axis, and the numerical value of this tension is equal to the force due to the action of this pressure on a circle whose diameter is equal to the conjugate axis of the ellipse.

    0
    0
  • The resultant of the internal pressure and the surface-tension is equivalent to a pressure along the axis equal to that due to a pressure p acting on a circle whose diameter is the conjugate axis of the hyperbola.

    0
    0
  • The resultant force on C will therefore tend to oppose the displacement and to bring C back to its original Nouvelle the'orie de l'action capillaire (1831).

    0
    0
  • It is also stable as regards displacements transverse to the axis, for the film is in a state of tension, and any lateral displacement of its middle parts would produce a resultant force tending to restore the film to its original position.

    0
    0
  • But if the length of the cylindric film is greater than its circumference, and if we suppose the disk C to be placed midway between A and B, and to be moved towards A, the pressure on the side next A will diminish, and that on the side next B will increase, so that the resultant force will tend to increase the displacement, and the equilibrium of the disk C is therefore unstable.

    0
    0
  • This very naturally resulted in a too frequent substitution of clerical concubinage for marriage; and the resultant evils form one of the commonest themes of complaint in church councils of the later middle ages.'

    0
    0
  • In the case of a beautiful object the resultant pleasure borrows its specific quality from the presence of determinations essentially objective in their nature, though not reducible to the categories of science.

    0
    0
  • Thus, though the average value of agricultural land increased by 60% between 1870 and 1900, the position of the peasantry is far from satisfactory, and the resultant discontent was the chief cause of the agrarian rising in 1907.

    0
    0
  • Then a is the resultant horizontal pressure with an over turning moment of (I) Equating the moment of resistance (2) to the overturning moment (I), we have pxzd =d3 3 6 and x =?2 p ..

    0
    0
  • The resultant profile is of the kind shown in fig.

    0
    0
  • After Rankine, a French engineer, Bouvier, gave the ratio of the maximum stress in a dam to the maximum vertical stress as 1 to the cosine squared of the angle between the vertical and the resultant which, in dams of the usual form, is about as 13 is to 9.

    0
    0
  • On examining the diagram it will be observed that the maximum compressive stresses are parallel to and near to the down stream face of the section, which values are approximately equal to the maximum value of the vertical stress determined by the law of uniformly varying stress divided by the cosine squared of the angle between the vertical and the resultant.

    0
    0
  • The resultant legislature (at Pawnee, later at Shawnee Mission) adopted the laws of Missouri almost en bloc, made it a felony to utter a word against slavery, made extreme pro-slavery views a qualification for office, declared death the penalty for aiding a slave to escape, and in general repudiated liberty for its opponents., The radical free-state men thereupon began the importation of rifles.

    0
    0
  • the elimination from the two equations of y (or x) gives for x (or y) an equation of a certain order, say the resultant equation; and then to each value of x (or y) satisfying this equation there corresponds in general a single value of y (or x), and consequently a single point of intersection; the number of intersections is thus equal to the order of the resultant equation in x (or y).

    0
    0
  • Supposing that the two curves are of the orders m, n, respectively, then the order of the resultant equation is in general and at most = mn; in particular, if the curve of the order n is an arbitrary line (n= 1), then the order of the resultant equation is = m; and the curve of the order m meets therefore the line in m points.

    0
    0
  • But the resultant equation may have all or any of its roots imaginary, and it is thus not always that there are m real intersections.

    0
    0
  • The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m; for instance, taking m= 2, if the hyperbola xy - 1= o be cut by the line y=0, the resultant equation in x is Ox- 1 = o, and there is apparently only the intersection (x 110, y =0); but the theorem is, in fact, true for every line whatever: a curve of the order in meets every line whatever in precisely m points.

    0
    0
  • resultant action may be looked upon as giving a mixture meat by of equal volumes of hydrogen and carbon monoxide, both meatretted of which are inflammable but non-luminous gases.

    0
    0
  • Conduct was regarded as the result of interaction between character and environment; or it was asserted to be the resultant effect of a struggle between motives in which the strongest prevailed.

    0
    0
  • (i) the general power of choosing among different alternatives of action without a motive, or against the resultant force of conflicting motives; (2) the power of choice between the promptings of reason and those of appetites (or other non-rational impulses) when the latter conflict with reason; (3) merely the quality of acting rationally in spite of conflicting impulses, however strong, the non posse peccare of the medieval theologians.

    0
    0
  • The periodic variations can be represented algebraically as the resultant of a series of harmonic motions in the following way: Let L be an angle which is increasing uniformly with the time, and let n be its rate of increase.

    0
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  • This is not the direction of gravity proper, or of the earth's attraction, but the resultant of this attraction combined with the centrifugal force due to the earth's rotation on its axis.

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  • 1 The problem he attempted to solve was so to combine uniform circular movements as to produce the resultant effects actually observed.

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  • These compounds are brought into solution by means of polysulphides of the alkali metals and the resultant liquor run into the cathode compartment of a bath, which is divided by diaphragms into a series of anode and cathode chambers; the anode divisions being closed and gas-tight, and containing carbon or platinum electrodes.

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  • The resultant of these is = 2a cos 2 (k 2 - k i)z cos {nt - 1(k2 -Fk2)z}, = 2a sin 2 (k 2 - ki)z cos {nt - z (k i + k2)z}, which shows that for any fixed value of z the light is plane polarized in a plane making an angle 1(k 2 - ki)z = ir(X i - X7 1)z, with the initial plane of polarization, X 1 and being the wave-lengths of the circular components of the same frequency.

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  • When the plates are of equal thickness, their combined effect is nil, but by adjusting the second, a rotation in the one or the other direction may be introduced, a scale attached to one prism and a vernier to the other giving the thickness of the resultant quartz plate.

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  • - It is instructive to consider the distribution of potential in a thermoelectric circuit, and its relation to the resultant E.M.F.

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  • If the circuit is cut at this point, there remains a difference of potential E =1184 microvolts, the resultant E.M.F.

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  • Far from having impressed its own direction on the orientation of the chain at large, this crest is merely the resultant of secondary agencies by which the primitive mass has been eroded and lessened in bulk, and though its importance from a hydrographic point of view is still considerable, its geological significance is practically nil.

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  • Fecundation would under such conditions be impossible, and without this the eggs of a resultant queen will produce nothing but drones.

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  • Microscopic sections of some of the more coherent clays and shales may be prepared by saturating them with Canada balsam by long boiling, and slicing the resultant mass in the same manner as one of the harder rocks.

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  • asymptomatic for many years and the resultant cyst can grow very large, eventually involving several liters of fluid.

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  • The use of a secondary azide was successful, although the stereoselectivity of the reduction of the resultant iminium ion was low.

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  • Barrie, duly chastened, took up position to defend the resultant freekick which came to naught.

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  • With Miedzinski passing 16-year-old Danish debutant Jonas Raun, the resultant 7-2 advantage saw Swindon haul themselves back into the match.

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  • dilatation of the pupils and resultant dimness of vision.

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  • If the egg is not fertilized, however, resultant hormonal changes cause the endometrium to slip away and menstruation begins.

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  • The resultant write-down of fixed assets and stocks has been included in operating costs but has been treated as exceptional.

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  • Furthermore, the resultant particles, called composite fermions, interact with each other very little.

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  • The resultant gene reshuffling may be responsible for a variety of chronic diseases (see " dynamic genomics ", this series ).

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  • This drop in sales causes already hard-pressed industry to suffer a drop in sales, with resultant lay-offs, redundancies and business failures.

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  • TMD said that inter parties work was becoming more complicated at an earlier stage with a resultant increase in correspondence.

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  • The wavelength associated with the vector resultant of these three orthogonal propagation constants is just the free space wavelength lambda.

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  • This lecture describes the action of wind upon desert sediments and the resultant desert landforms.

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  • Vodka: Got used in the production of a strawberry liqueur that had a resultant half-life of about 25 minutes.

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  • To address problems that arise in the resultant optimization we introduce a technique called spherical normalization that preconditions the Hessian matrix.

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  • mauls from the resultant line-outs.

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  • Both Tresarrieu and Janniro hit the deck in the resultant melee and for a short time, there was concern for prostrate Adams.

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  • One would expect this random method of notation to be discordant, however the resultant music is surprisingly reminiscent of classical piano minuets.

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  • Oxfam warned that such obstinacy from the EU and the resultant standoff between major trading powers would ultimately harm developing countries.

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  • The resultant magnetic field causes temporary optical illusions and distortions.

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  • The resultant prolonged standing ovation is a memory that will stay with United fans for many years to come.

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  • The final probability of an event is then obtained by the sum of the squares of the two numbers describing the resultant probability amplitude.

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  • Most samples exhibited a reduced sheet resistivity resultant from film curing over the duration of the test.

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  • resultant of the two forces playing against each other that makes good work.

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  • resultant of a number of forces.

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  • resultant of two vectors using the vector triangle.

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  • resultant free-kick, taken by Michael Rose on the home right, County grabbed the lead.

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  • resultant waveforms of V L and V R.

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  • resultant disruption to students ' studies.

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  • resultant free kick Liverpool won a corner.

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  • resultant penalty to seal another draw at York Road.

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  • resultant corner, Barrett was well place to head off the line.

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  • Falling nearby with the resultant loud explosion and debris scattered all over.

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  • Mark Lock then picked up from the back of the resultant scrum and was again held up.

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  • The resultant prolonged standing ovation is a memory that will stay with United fans for many years to come.

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  • Chain Bearing stress The bearing stress is the resultant tensile stress / the bearing area B a.

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  • For lagged profiles that extend on either side of measured cells, the resultant matrix will be approximately symmetrical along the leading diagonal.

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  • Reduced serum thyroxine led to an increased release of TSH with resultant follicular cell hypertrophy in the thyroid glands of rats.

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  • tickle did it using the Multiface One, tickling up the resultant screen using Artist II.

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  • Follow immediately behind with a wet vac to lift the resultant slurry.

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  • vinification techniques ensures the resultant wines are packed with fruit and thoroughly modern in style.

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  • vitiate the resultant exercise of discretion whether to make either such order.

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  • The resultant waveform is stored in AV to be viewed.

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  • If the primary wave be represented by = e-ikx the component rotations in the secondary wave are '1'3= P (- AN y) N r2 ' cwi= r x D y N 'y)' lw2=P (- AD + 6,N z2 - x2 ' D r N r2 where ik3T e-ikr _ P - 4 r The expression for the resultant rotation in the general case would be rather complicated, and is not needed for our purpose.

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  • Accordingly, if E be the energy of the primary wave, dE 87-2n (D' - D) 2 T2 E dx 3 D2%4 ' whence E = Eoe-hx (II) where h = 8?r 2 n (D' - D)2T2 3 D2 x 4, (12) If we had a sufficiently complete expression for the scattered light, we might investigate (12) somewhat more directly by considering the resultant of the primary vibration and of the secondary vibrations which travel in the same direction.

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  • Moreover, if OP = r, and AO=x, then r 2 =x 2 + p2, and pdp=rdr. The resultant at 0 of all the secondary vibrations which issue from the stratum dx is by (3), with sin ¢ equal to unity, ndx f ?

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  • cos(21r/X)(bt - x), and the resultant will then represent the whole actual disturbance at 0 as modified by the particles in the stratum dx.

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  • The incident light being white, the intensity of the component colours scattered in this direction varies as the inverse eighth power of the wave-length, so that the resultant light is a rich blue.

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  • Through the resultant scarcity of labour, much land fell out of cultivation.

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  • The radiations interfere in an optical sense of the word, and in some directions reinforce each other and in other directions neutralize each other, so making the resultant radiation greater in some directions than others.

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  • The main branches of the resultant " tree " may be rendered as follows: [[Coraciomorphae Odontolcae..Colymbo-+Pelargoalectoromorphae..Ratitae Morphae Morphae ' 'Neornithes]] The Odontolcae seem to be an early specialized offshoot of the Colymbo-Pelargomorphous brigade, while the Ratitae represent a number of side branches of early Alectoromorphae.

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  • It is also the custom to balance a proportion of the reciprocating masses by balance weights placed between the spokes of the wheels, and the actual balance weight seen in a driving-wheel is the resultant of the separate weights required for the balancing of the revolving parts and the reciprocating parts.

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  • Another of Roberval's discoveries was a very general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.

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  • He treated the resultant electric force at any point as analogous to the flux of heat from sources distributed in the same manner as the supposed electric particles.

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  • He considers that in its earliest origins Christian faith and the methods of Greek thought were so closely intermingled that much that is not essential to Christianity found its way into the resultant system.

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  • The general character of the country, resultant on these conditions, varies according to elevation and latitude.

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  • temperance and the resultant health and vigour).

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  • There are two gravitational fields which sometimes reinforce and at other times diminish each other and the effect is always a resultant one.

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  • When the solutions of two substances are mixed, similar considerations to those given above enable us to calculate the resultant changes in dissociation.

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  • R is a function of the coefficients which is called the " resultant " or " eliminant " of the k equations, and the process by which it is obtained is termed " elimination."

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  • If al, a2, ...a, n be the roots of f=o, (1, R2, -Ai the roots of 0=o, the condition that some root of 0 =o may qq cause f to vanish is clearly R s, 5 =f (01)f (N2) � �;f (Nn) = 0; so that Rf,q5 is the resultant of f and and expressed as a function of the roots, it is of degree m in each root 13, and of degree n in each root a, and also a symmetric function alike of the roots a and of the roots 1 3; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of f, and homogeneous and of degree m in the coefficients of 4..

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  • This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms.

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  • the sum of the suffixes in each term of the resultant is equal to mn.

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  • Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.

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  • +al pxp = 0, a21x1 +a22x2 + � � � +a2pxp = 0, aplxl+ap2x2+...+appxp = 0, be the system the condition is, in determinant form, (alla22...app) = 0; in fact the determinant is the resultant of the equations.

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  • Forming the resultant of these equations we evidently obtain the resultant of f and 4,.

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  • Thus to obtain the resultant of aox 3 +a i x 2 +a 2 x+a 3, 4, =box2+bix+b2 we assume the identity (Box+Bi)(aox 3 +aix 2 +a2x+a3) = (Aox 2 +Aix+ A 2) (box2+bix+b2), and derive the linear equations Boa ° - Ac b o = 0, Boa t +B i ao - A 0 b 1 - A 1 bo =0, Boa t +B 1 a 1 - A0b2 - A1b1-A2b° = 0, Boa3+Bla2 - A l b 2 -A 2 b 1 =0, B 1 a 3 - A 2 b 2 =0, = = (y l, y2,...ynl `x1, x2,...xnl for brevity.

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  • and by elimination we obtain the resultant ao 0 bo 0 0 al ao b1 bo 0 a 2 a i b 2 b 1 bo a numerical factor being disregarded.

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  • Taking the same example as before the process leads to the system of equations acx 4 +alx 3 +a2x 2 +a3x =0, aox 3 +a1x 2 +a2x+a 3 = 0, box +bix -1-b2x =0, box' +b i x 2 -{-h 2 x = 0, box + b i x + b:: = 0, whence by elimination the resultant a 0 a 1 a 2 a 3 0 0 a 0 a 1 a 2 a3 bo b 1 b 2 0 0 0 bo b 1 b 2000 bo b 1 b2 which reads by columns as the former determinant reads by rows, and is therefore identical with the former.

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  • Bezout's method gives the resultant in the form of a determinant of order m or n, according as m is n.

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  • He first divides by the factor x -x', reducing it to the degree m - I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers xo, x, - of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m.

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  • Put (aox 3 -}-a l x 2 +a 2 x +a 3) (box' +b1x'+b2) - (aox'3+aix'2+a2x'+a3) (box' + bix + b2) = 0; after division by x-x the three equations are formed aobcx 2 = aobix+aob2 =0, aobix 2 + (aob2+a1b1-a2bo) x +alb2 -a3bo = 0, aob2x 2 +(a02-a3bo)x+a2b2-a3b1 =0 and thence the resultant aobo ao aob2 aob 1 aob2+a1b1-a2bo alb2-a3b0 aob 2 a1b2 - a 3 bo a2b2 - a3b1 which is a symmetrical determinant.

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  • There is no difficulty in expressing the resultant by the method of symmetric functions.

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  • = 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, corn bined with the chosen values of y and z, yields a system of values which satisfies both equations.

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  • Hence this product is the required resultant of the three equations.

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  • Hence, finally, the resultant is expressed in terms of the coefficients of the three equations, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees np and pm in the coefficients of the first and second equations respectively.

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  • The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner.

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  • Cayley, however, has shown that, whatever be the degrees of the three equations, it is possible to represent the resultant as the quotient of two determinants (Salmon, l.c. p. 89).

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  • Discriminants.-The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables.

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  • It is the resultant of k polynomials each of degree m-I, and thus contains the coefficients of each form to the degree (m-I)'-1; hence the total degrees in the coefficients of the k forms is, by addition, k (m - 1) k - 1; it may further be shown that the weight of each term of the resultant is constant and equal to m(m-I) - (Salmon, l.c. p. loo).

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  • This can be seen at once because the factor in question being once repeated in both differentials, the resultant of the latter must vanish.

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  • = = o, and have in consequence a vanishing resultant.

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  • af Expression in Terms of Roots.-Since x+y y =mf, if we take cx any root x 3, y1, ofand substitute in mf we must obtain, y 1 C) zaZ1 �; hence the resultant of and f is, disregarding numerical factors, y,y2...y,,.

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  • (x y m - x m y), ar _ y1(x y 2 - and substituting in the latter any root of f and forming the product, we find the resultant of f and d, viz.

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  • The discriminant of the product of two forms is equal to the product of their discriminants multiplied by the square of their resultant.

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  • Schlafli 1 this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables.

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  • This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a l, a2; b l, b 2 are alternative, are ultimately put equal to one another and (ab) vanishes.

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  • In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant.

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  • The existence of such forms seems to have been brought to Sylvester's notice by observation of the fact that the resultant of of and b must be a factor of the resultant of Xax+ 12 by and X'a +tA2 for a common factor of the first pair must be also a common factor so we obtain P: = of the second pair; so that the condition for the existence of such common factor must be the same in the two cases.

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  • Remark.-The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C = o, p is the common factor of i and j.

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  • The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.

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  • 1 A2B' Where The Denominator Factors Indicate The Forms Themselves, Their Jacobian, The Invariant Of The Quadratic And Their Resultant; Connected, As Shown By The Numerator, By A Syzygy Of Degreesorder (2, 2; 2).

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  • The process of magnetization consists in turning round the molecules by the application of magnetic force, so that their north poles may all point more or less approximately in the direction of the force; thus the body as a whole becomes a magnet which is merely the resultant of an immense number of molecular magnets.

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  • A magnet attached to a cork and [[[Terminology And Principles]] floated upon water will set itself with its axis in the magnetic meridian, but it will be drawn neither northward nor southward; the forces acting upon the two poles have therefore no horizontal resultant.

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  • And again if a piece of steel is weighed in a delicate balance before and after magnetization, no change whatever in its weight can be detected; there is consequently no upward or downward resultant force due to magnetization; the contrary parallel forces acting upon the poles of the magnet are equal, constituting a couple, which may tend to turn the body, but not to propel it.

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  • terminate outside the magnet or inside, have a resultant, equal to the sum of the forces and parallel to their direction, acting at a certain point N.

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  • Similarly, the forces acting in the opposite direction on the negative poles of the filaments have a resultant at another point S, which is called the south or negative pole.

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  • The line through the given point along which the potential decreases most rapidly is the direction of the resultant magnetic force, and the rate of decrease of the potential in any direction is equal to the component of the force in that direction.

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  • If V denote the potential, F the resultant force, X, Y, Z, its components parallel to the co-ordinate axes and n the line along which the force is directed, then - sn = F, b?= X, - Sy = Y, -s Surfaces for which the potential is constant are called equipotential surfaces.

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  • The resultant magnetic force at every point of such a surface is in the direction of the normal (n) to the surface; every line of force therefore cuts the equipotential surfaces at right angles.

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  • For the resultant force at P, F=-VF r 2.

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  • Since 7ra'I is the moment of the sphere (=volume X magnetization), it appears from (10) that the magnetized sphere produces the same external effect as a very small magnet of equal moment placed at its centre and magnetized in the same direction; the resultant force therefore is the same as in (14).

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  • The resultant magnetic field, therefore, is compounded of two fields, the one being due to the poles, and the other to the external causes which would be operative in the absence of the magnetized metal.

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  • The intensity (at any point) of the field due to the magnetization may be denoted by H i, that of the external field by Ho, and that of the resultant field by H.

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  • Magnetization is usually regarded as the direct effect of the resultant magnetic force, which is therefore often termed the magnetizing force.

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  • Demagnetizing Force.-It has already been mentioned that when a ferromagnetic body is placed in a magnetic field, the resultant magnetic force H, at a point within the body, is compounded of the force H o, due to the external field, and of another force, Hi, arising from the induced magnetization of the body.

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  • Equations (33) and (34) show that when, as is generally the case with ferromagnetic substances, the value of is considerable, the resultant magnetic force is only a small fraction of the external force, while the numerical value of the induction is approximately three times that of the external force, and nearly independent of the permeability.

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  • The action of a hollow magnetized shell on a point inside it is always opposed to that of the external magnetizing force, 6 the resultant interior field being therefore weaker than the field outside.

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  • principal axes of the crystal, the actual magnetization will be the resultant of the three magnetizations along the axes.

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  • If two magnets having moments M, M' are arranged at right angles to each other upon a horizontal support which is free to rotate, their resultant R will set itself in the magnetic meridian.

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  • 5), we have for the resultant force at P R = - 2cosO X m/d' = - elm/d 3 = - M/d3.

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  • The direction of the resultant field of force will then make, with that of H E, an angle 0, such that Hp/H E tan 0, and the suspended needle will be deflected through the same angle.

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  • Then, since Hp, the force at M due to m and - m, is the resultant of d, and - m2, we have (L Hp m d d2 2ml H = d3, the direction being parallel to AB.

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  • The wire is subject to two superposed magnetizations, the one longitudinal, the other circular, due to the current traversing the wire; the resultant magnetization is consequently in the direction of a screw or spiral round the wire, which will be right-handed or left-handed according as the relation between the two magnetizations is right-handed or left-handed; the magnetic expansion or contraction of the metal along the spiral lines of magnetization produces the Wiedemann twist.

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  • Weber therefore supposed each molecule to be acted on by a force tending to preserve it in its original direction, the position actually assumed by the axis being in the direction of the resultant of this hypothetical force and the applied magnetizing force.

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  • many different directions, so that there is no resultant magnetic moment.

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  • If now a gradually increasing magnetizing force is applied, the needles at first undergo a stable deflection, giving to the group a small resultant moment which increases uniformly with the force; and if the current is interrupted while the force is still weak, the needles merely return to their initial positions.

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  • The rearrangement is completed within a comparatively small range of magnetizing force, a rapid increase of the resultant moment being thus brought about.

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  • When the field is removed, many of the newly formed combinations are but slightly disturbed, and the group may consequently retain a considerable resultant moment.

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  • This corresponds to the second stage of magnetization, in which the susceptibility is large and permanent magnetization is set up. A still stronger magnetizing force has little effect except in causing the direction of the needles to approach still more nearly to that of the field; if the force were infinite, every member of the group ‘ would have exactly the same direction and the greatest possible resultant moment would be reached; this illustrates " magnetic saturation " - the condition approached in the third stage of magnetization.

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  • If the structure of the molecule is so perfectly symmetrical that, in the absence of any external field, the resultant magnetic moment of the circulating electrons is zero, then the application of a field, by accelerating the right-handed (negative) revolutions, and retarding those which are left-handed, will induce in the substance a resultant magnetization opposite in direction to the field itself; a body composed of such symmetrical molecules is therefore diamagnetic. If however the structure of the molecule is such that the electrons revolving around its atoms do not exactly cancel one another's effects, the molecule constitutes a little magnet, which under the influence of an external field will tend to set itself with its axis parallel to the field.

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  • The phase of the resultant is midway between those of the extreme elements, that is to say, a quarter of a period behind that due to the element at the centre of the circle.

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  • The middle element alone contributes without deduction; the effect of every other must be found by introduction of a resolving factor, equal to cos 0, if 0 represent the difference of phase between this element and the resultant.

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  • Accordingly, the amplitude of the resultant will be less than if all its components had the same phase, in the ratio +17r -17r or 2: 7.

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  • The process of augmenting the resultant illumination at a particular point by stopping some of the secondary rays may be carried much further (Soret, Pogg.

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  • When the difference of phase amounts to A, we may expect the resultant illumination to be very much reduced.

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  • The phase of the resultant amplitude is the same as that due to the central secondary wave, and the discrepancies of phase among the components reduce the amplitude in the proportion l ` dri): 3?

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  • Since at these points the resultant due to the whole aperture is zero, any two portions into which the whole may be divided must give equal and opposite resultants.

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  • The effect of each of the elements of the grating is then the same; and, unless this vanishes on account of a particular adjustment of the ratio a: d, the resultant amplitude becomes comparatively very great.

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  • The phase of the resultant effect is by symmetry that of the component which comes from the middle of a.

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  • The fact that the other components have phases differing from this by amounts ranging between tam 2 r/(a+d) causes the resultant amplitude to be less than for the central image (where there is complete phase agreement).

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  • The vibrations corresponding to the two parts are precisely antagonistic, since if both were operative the resultant would be zero.

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  • Hence, in accordance with the rule for compounding vector quantities, the resultant vibration at B, due to any finite part of the primary wave, is represented in amplitude and phase by the chord joining the extremities of the corresponding arc (U2-0.1).

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  • If the resultant rotation be n, we have TZ iJ (x 2 -{-y 2) de ikr TZsin4 d e ikr 2 r ' dr (r !

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  • that at any point the tangent to the hodograph is parallel to the direction, and the velocity in the hodograph equal to the magnitude of the resultant acceleration at the corresponding point of the orbit.

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  • Phil.): - Let x, y, z be the coordinates of P in the orbit,, r t, those of the corresponding point T in the hodograph, then dx dy _ dz c= ' 71 - a' - at therefore Also, if s be the arc of the hodograph, ds = v = V V1 1) j dt + (dt2) dt Equation (1) shows that the tangent to the hodograph is parallel to the line of resultant acceleration, and (2) that the velocity in the hodograph is equal to the acceleration.

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  • For if the body is removed, and replaced by the fluid as at first, this fluid is in equilibrium under its own weight and the thrust of the surrounding fluid, which must be equal and opposite, and the surrounding fluid acts in the same manner when the body replaces the displaced fluid again; so that the resultant thrust of the fluid acts vertically upward through the centre of gravity of the fluid displaced, and is equal to the weight.

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  • The resultant vertical thrust on any portion of a curved surface exposed to the pressure of a fluid at rest under gravity is the weight of fluid cut out by vertical lines drawn round the boundary of the curved surface.

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  • The resultant horizontal thrust in any direction is obtained by drawing parallel horizontal lines round the boundary, and intersecting a plane perpendicular to their direction in a plane curve; and then investigating the thrust on this plane area, which will be the same as on the curved surface.

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  • In casting a thin hollow object like a bell, it will be seen that the resultant upward thrust on the mould may be many times greater than the weight of metal; many a curious experiment has been devised to illustrate this property and classed as a hydrostatic paradox (Boyle, Hydrostatical Paradoxes, 1666).

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  • The resultant force is therefore in the direction of the steepest pressure-gradient, and this is normal to the surface of equal pressure; for equilibrium to exist in a fluid the lines of force must therefore be capable of being cut orthogonally by a system of surfaces, which will be surfaces of equal pressure.

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  • - A plane area exposed to fluid pressure on one Side experiences a single resultant thrust, the integrated pressure over the area, acting through a definite point called the centre of pressure (C.P.) of the area.

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  • Thus if the plane is normal to Or, the resultant thrust R =f fpdxdy, (r) and the co-ordinates x, y of the C.P. are given by xR = f f xpdxdy, yR = f f ypdxdy.

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  • tan 0, so that the resultant righting couple is W.GM.

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  • Putting (12) a vortex line is defined to be such that the tangent is in the direction of w, the resultant of, n, called the components of molecular rotation.

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  • The osculating plane of a stream line in steady motion contains the resultant acceleration, the direction ratios of which are du du, du d i g d g 2 _ dH dx +v dy + dz - 2v?

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  • If other vortices are present, any one may be supposed to move with the velocity due to the others, the resultant stream function being = gy m log r =log IIrm; (9) the path of a vortex is obtained by equating the value of 1P at the vortex to a constant, omitting the rm of the vortex itself.

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  • When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (U 2 + V2)b2a a2U+Vi +b a b a2 U z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (U b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?

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  • Any circular filament can be started from rest by the application of a circumferential impulse 7rpmdr at each end of a diameter; so that a mechanism attached to the cylinders, which can set up a uniform distributed impulse rpm across the two parts of a diameter in the liquid, will generate the vortex motion, and react on the cylinder with an impulse couple-pmira 2 and pm7rb 2, having resultant pm7r(b 2 -a 2), and this couple is infinite when b = oo, as the angular momentum of the vortex is infinite.

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  • Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Ua 2 r2 cos 20 +mr i sin 0) (Ua 2 r 2 sin 2 0+mr 1 cos 0) + pdy (- U+Ua 2 r 2 cos 2 0 +mr1 sin 0) (Ua 2 r 2 sin 2 0 -mr 1 cos 0) =2pdymUr '(cos 0 -a 2 r 2 cos 30), (8) and with b tan r =b sec this is 2pmUdo(i -a 2 b2 cos 30 cos 0), (9) and integrating between the limits 0 = 27r, the resultant, as before, is 27rpmU.

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  • having a resultant in the direction PO, where P is the intersection of an ellipse n with the hyperbola 13; and with this velocity the ellipse n can be swimming in the liquid, without distortion for an instant.

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  • The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin Olga, and its thrust is 27rpmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r=a with velocity U reversed, and the cylinder is surrounded by a vortex.

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  • If at any points of a vortex line the cross-section ABC, A'B'C' is drawn of the vortex filament, joined by the vortex line AA', then, since the flow in AA' is taken in opposite directions in the complete circuit ABC AA'B'C' A'A, the resultant flow in AA' cancels, and the circulation in ABC, A'B'C' is the same; this is expressed by saying that at all points of a vortex filament wa is constant where a is the cross-section of the filament and w the resultant spin (W.

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  • The impulse required to set up the motion in liquid of density p i the resultant of an impulsive pressure p4) over the surface S of th ellipsoid, and is therefore ffp4ldS = p4GoffxldS =p 40 (volume of the ellipsoid) =4)oW', (23) where W' denotes the weight of liquid displaced.

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  • These equations are proved by taking a line fixed in space, whose direction cosines are 1, then dt=mR-nQ,' d'-t = nP =lQ-mP. (5) If P denotes the resultant linear impulse or momentum in this direction P =lxl+mx2+nx3, ' dP dt xl+, d y t x2' x3 +1 dtl dt 2 +n dt3, =1 ('+m (dt2-x3P+x1R) ' +n ('-x1Q-{-x2P) ' '= IX +mY+nZ, / (7) for all values of 1, Next, taking a fixed origin and axes parallel to Ox, Oy, Oz through 0, and denoting by x, y, z the coordinates of 0, and by G the component angular momentum about 1"2 in the direction (1, G =1(yi-x2z+x3y) m 2-+xlz) n(y(y 3x 1 x3x y + x 2 x) (8) Differentiating with respect to t, and afterwards moving the fixed.

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  • are the components of a constant vector having a fixed direction; while (4) shows that the vector resultant of y, y, y moves as if subject to a couple of components x Wx V, x Ux W, x V-x U, (Io) and the resultant couple is therefore perpendicular to F, the resultant of x, x, x, so that the component along OF is constant, as expressed by (iii).

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  • In the absence of a medium the inertia of the body to transtion is the same in all directions, and is measured by the (3) But the change of the resultant momentum F of the medium as.

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  • Consider a submarine boat or airship moving freely with the direction of the resultant momentum horizontal, and the axis at a slight inclination 0.

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  • For the body alone the resultant of the components of momentum W V -cos andW V sin 0 is W V -sec. lb, acting along 00', and so is unaltered.

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  • the moment of inertia of the body about the axis, denoted by But if is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k', If the shot is spinning about its axis with angular velocity p, and is precessing steadily at a rate about a line parallel to the resultant momentum F at an angle 0, the velocity of the vector of angular momentum, as in the case of a top, is C i pµ sin 0- C2µ 2 sin 0 cos 0; (4) and equating this to the impressed couple (multiplied by g), that is, to gN = (c 1 -c 2)c2u 2 tan 0, (5) and dividing out sin 0, which equated to zero would imply perfect centring, we obtain C21 2 cos 0- (c 2 -c 1)c2u 2 sec 0 =o.

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  • The body is held fixed, and the reactio of the mechanism and the resultant of the impulsive pressure on th surface are a measure of the impulse, linear,, , and angula A, µ, v, required to start the circulation.

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  • The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by the resultant axial linear momentum of the circulation.

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  • If a small conducting body is charged with Q electrostatic units of electricity, and placed in any electric field at a point where the electric force has a value E, it will be subject to a mechanical force equal to QE dynes, tending to move it in the direction of the resultant electric force.

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  • We may otherwise define it by saying that a line of electric force is a line so drawn in a field of electric force that its direction coincides at every point with the resultant electric force at that point.

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  • It can be shown that the resultant electric force normal to the surface at a point just outside a conductor is 1 See Maxwell, Elementary Treatise on Electricity (Oxford, 1881), P. 47.

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  • Hence we see that if the whole surface of the sphere is divided into pairs of elements by cones described through any interior point, the resultant force at that point must consist of the sum of pairs of equal and opposite forces, and is therefore zero.

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  • Since then they are all charged with the same quantity of electricity, and the total over all potential difference V is the sum of each of the individual potential differences V1, V2, V3, &c., we have Q=C I V I =C 2 V 2 =C 3 V 3 =&c., and V=V1-FV2+V3+&c. The resultant capacity is C = Q/V, and C= I/(I/C1 +I /C2+1/C3+&c) = I/Z(I /C) (15).

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  • These rules provide means for calculating the resultant capacity when any number of condensers are joined up in any way.

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  • The resultant electric force E at that point is then obtained by differentiating V, since E = - dV/dx, and E is in the direction in which V diminishes fastest.

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  • In any case, therefore, in which we can sum up the elementary potentials at any point we can calculate the resultant electric force at the same point.

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  • At each point in the field the electric force can have but one resultant value.

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  • At each point on this surface the resultant force has a certain value, and a certain direction inclined at an angle 0 to the normal to the selected surface at that point.

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  • Let E be the resultant electric force at any point in the field.

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  • Then the potential at any point P in this ideal plane PO is equal to q/AP-q/BP=0, whilst the resultant force at P due to the two point charges is 2gAO/AP 3, and is parallel to AB or normal to PO.

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  • The resultant force due to these two pointcharges must then be in the direction CP, and its value E is the vector sum of the two forces along AP and BP due to the two point-charges.

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  • If, therefore, the walls of the enclosure held the gas that is directly in contact with them, this equilibrium would be the actual state of affairs; and it would follow from the principle of Archimedes that, when extraneous forces such as gravity are not considered, the gas would exert no resultant force on any body immersed in it.

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  • When a shaft is driven by means of gearing the driving torque is measured by the product of the resultant pressure P acting between the wheel teeth and the radius of the pitch circle of the wheel fixed to the shaft.

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  • (5) If c is the resultant velocity of a molecule, so that c 2 =u2+v2+w2, it is readily found from formula (4) that the number of molecules of the first kind of which the resultant velocity lies between c and c+dc is 4lrs1,l (h 3 rn 3 17r 3)e hmc2 c 2 dc. (6) These formulae express the " law of distribution of velocities " in the normal state: the law is often called Maxwell's Law of Distribution.

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  • But it can be shown that from the aggregation of these separate short motions the particle ought to have a resultant motion, described with an average velocity which, although much smaller than 2 mm.

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  • Even the resultant of mechanical forces refuses to resolve itself into its constituents.

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  • In the " resultant " there is a new direction, and with it a new quality the component forces of which no analysis can discover.'

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  • They adapted to these conditions some of the methods for managing local affairs with which they had been familiar in England, and called the resultant institution a town.

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  • HH' = 60 X 60' the resultant angle of deflection is HFH', and this can be determined by the same formula a = h X1200 X 3, but in this case h = HH' = R R 60 60 R X 3600 a _ = i' so that if the sight is inclined to the left I° it will R X 3600 give 1' deflection for every degree of elevation.

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  • the resultant with amplitude I/-/2 that of (1).

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  • Another movement of 4A in each direction gives (3) with resultant a straight line, and so on for (4) and (5).

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  • Since the resultant is zero, mU 2 sin 4) - T sin ct.

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  • If the two interfering waves, being still of same length X, be in opposite phases, or sõ that one is in advance of the other by 2X, and consequently one produces in the air the opposite state of motion to the other, then the resultant wave is one of the same length X, but the excursions of the particles are decreased, being the difference between those due to the component waves as in fig.

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  • If we measure the time from an instant at which the two are in the same phase the resultant disturbance is y=a sin i t+a sin 27rn2t =2a cos ir(n i - n 2)t sin ir(nl-t-n2)t, which may be regarded as a harmonic disturbance of frequency (ni+n2)/2 but with amplitude 2a cos 7r(n i - n 2)t slowly varying with the time.

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  • Sovereignty is a resultant of many forces.

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  • When girders form the superstructure, the resultant pressure on the piers or abutments is vertical, and the dimensions of these are simply regulated by the sufficiency to bear this vertical load.

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  • When arches form the superstructure, the abutment must be so designed as to transmit the resultant thrust to the foundation in a safe direction, and so distributed that no part may be unduly compressed.

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  • When the resultant pressure is not vertical on the piers these must be constructed to meet the inclined pressure.

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  • beyond C, the shear at C will probably be greatest when W2 is at C. Let R be the resultant of the loads on the bridge when W 1 is at C. Then the reaction at B and shear at C is Rn/l.

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  • 46) be the section at which the greatest shear is required, and let the loads advance from the left till W I is at C. If R is the resultant of the loads then on the girder, the reaction at B and shear at C is Rn/l.

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  • 70, at a distance x from the vertex, the horizontal component of the resultant (tangent to the curve) will be unaltered; the vertical component V will be simply the sum of the loads between 0 and F, or wx.

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  • resultant tension along the chain at F, the vertical force V passing through the point D, and the horizontal tension at O; hence H: V = DC: FC = wx 2 /2 y: wx = x/2.

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  • The nature of the resultant concepts belongs to the great controversy between Nominalism, Realism and Conceptualism.

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  • The most fundamental experimental confirmation that the theory of the aether has received on the optical side in recent years has been the verification of Maxwell's proposition that radiation exerts mechanical force on a material system, on which it falls, which may be represented in all cases as the resultant of pressures operating along the rays, and of intensity equal at each point of free space to the density of radiant energy.

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  • 26-27; the length of the ministry was fixed, with some approach to certainty, at between two and three years, and here too the resultant date for the Crucifixion would be the Passover of A.D.

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  • Another question with which the textual critic of modern authors must be prepared to deal is the relative importance of different editions, each of which may have a prima facie claim to be considered authentic. Thus Shakespearean criticism must decide between the evidence of the first folio and the quartos: the critic of Shelley's poems must consider what weight is to be attached to the readings in the posthumous edition by Mrs Shelley, and in unpublished transcripts of various poems. Where there is great or complicated divergence between the editions, as in the case of Marlowe's Faustus, the production of a resultant text which may be relied upon to represent the ultimate intention of the author is well-nigh impossible.

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  • The umbrella analogy is similarly explained; the most efficient position being when the stick points along the resultant AD.

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  • A full description of its modern condition is therefore given under the heading Oviedo; the present article being confined to an account of its physical features, its history, and the resultant character of its inhabitants.

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  • Instead of the operation of superposing accelerations, we may compound the several forces acting on a particle by the parallelogram law (see Mechanics) into what may be called the resultant force, the total acceleration of the particle being the same as if this alone acted.

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  • What is done is to divide the resultant force due to gravitation into two components, one of which corresponds to this acceleration, while the other one is what is called the "weight" of the body.

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  • The distinction between these two classes is not sharp; though when the properties of the resultant are sensibly the sum of those of the pure components, as is nearly the case for a complex gas such as air, it is usual to class it as a mixture.

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  • When the properties of the resultant substance are different from those of the components and it is not a chemical compound we define it as a solution.

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  • By cooling the resultant solution through the range dT we recover the original state of the system.

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  • The fuel, wood or charcoal, which served both to heat and to deoxidize the ore, has so strong a carburizing action that it would turn some of the resultant metal into " natural steel," which differs from wrought iron only in containing so much carbon that it is relatively hard and brittle in its natural state, and that it becomes intensely hard when quenched from a red heat in water.

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  • In 1856 Bessemer not only invented his extraordinary process of making the heat developed by the rapid oxidation of the impurities in pig iron raise the temperature above the exalted melting-point of the resultant purified steel, but also made it widely known that this steel was a very valuable substance.

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  • Slag or Cinder, a characteristic component of wrought iron, which usually contains from 0.20 to 2.00% of it, is essentially a silicate of iron (ferrous silicate), and is present in wrought iron simply because this product is made by welding together pasty granules of iron in a molten bath of such slag, without ever melting the resultant mass or otherwise giving the envelopes of slag thus imprisoned a chance to escape completely.

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  • Part of the resultant carbonic acid is again deoxidized to carbonic oxide by the surrounding fuel, CO 2 + C = 2C03 and the carbonic oxide thus formed deoxidizes more iron oxide, &c. As indicated in fig.

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  • Moreover, the quality of the resultant steel depends upon the temperature of the process, and this in turn depends upon the proportion of silicon, the combustion of which is the chief source of the heat developed.

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  • In the basic Bessemer process, also, unforeseen variations in the siliconcontent are harmful, because the quantity of lime added should be just that needed to neutralize the resultant silica and the phosphoric acid and no more.

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  • In the Bessemer or " pneumatic" process, which indeed might be called the " fuel-less " process, molten pig iron is converted into steel by having its carbon, silicon and manganese, and often its phosphorus and sulphur, oxidized and thus removed by air forced through it in so many fine streams and hence so rapidly that the heat generated by the oxidation of these impurities suffices in and by itself, unaided by burning any other fuel, not only to keep the iron molten, but even to raise its temperature from a point initially but little above the melting point of cast iron, say 1150 to 1250° C., to one well above the melting point of the resultant steel, say i soo C. The " Bessemer converter " or " vessel " (fig.

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  • so powerful an acid as silica, then the phosphoric acid has so feeble a hold on the base in the slag that it is immediately redeoxidized by the carbon of the metal, or even by the iron itself, P 2 O 5 +5Fe = 2P+5FeO, and the resultant deoxidized phosphorus immediately recombines with the iron.

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  • - Silicon cannot here be used as the chief source of heat as it is in the acid Bessemer process, because most of the heat which its oxidation generates is consumed in heating the great quantities of lime needed for neutralizing the resultant silica.

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  • Further objections to the presence of silicon are that the resultant silica (1) corrodes the lining of the converter, (2) makes the slag froth so that it both throws much of the charge out and blocks up the nose of the converter, and (3) leads to rephosphorization.

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  • But Massenez and Richards, following the plan outlined by Pourcel in 1879, have found that even 3% of silicon is permissible if, by adding iron ore, the resultant silica is made into a fluid slag, and if this is removed in the early cool part of the process, when it attacks the lining of the converter but slightly.

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  • Manganese to the extent of 1.80% is desired as a means of preventing the resultant steel from being redshort, i.e.

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  • The proportion of pig to scrap used depends chiefly on the relative cost of these two materials, but sometimes in part also on the carbon-content which the resultant steel is to have.

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  • The ebullition from the formation of carbonic oxide puffs up the resultant phosphoric slag enough to make most of it run out of the furnace, thus both removing the phosphorus permanently from danger of being later deoxidized and returned to the steel, and partly freeing the bath of metal from the heat-insulating blanket of slag.

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  • Floating on top of the molten metal, it rapidly oxidizes its phosphorus, and the resultant phosphoric acid combines with the lime in the overlying slag as phosphate of lime.

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  • It is practically unattainable in the open-hearth furnace, because here the oxygen of the furnace atmosphere indirectly oxidizes the carbon of the metal which is kept boiling by the escape of the resultant carbonic oxide.

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  • The resultant of these two effects has not yet been well established; but it is probable that the strongest cast iron has a little more than 1% of carbon combined as cementite, so that its matrix is nearly equivalent to the strongest of the steels.

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