# How to use Resultant in a sentence

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

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

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

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

• 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).

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

• 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.

• 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.

• 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.

• 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.

• 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).

• 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.

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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

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

• 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.

• 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.

• 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.

• 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.

• 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.

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

• 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.

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

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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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?

• 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.

• 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.

• 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.

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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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).

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

• 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.

• 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.

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

• 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.

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

• 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.

• 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.

• 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.

• 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.

• 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.

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

• 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.

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

• 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.

• 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.

• Sovereignty is a resultant of many forces.

• 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.

• 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.

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

• 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.

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

• 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.

• 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.

• 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.

• 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.

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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

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

• 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.

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

• 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.

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

• 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.

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

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

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

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

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

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

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

• 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.

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

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

• 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.

• 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.

• 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.

• 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.

• 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.

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

• 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.

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

• 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.

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

• 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.

• 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.

• 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.

• 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.

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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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 -

• 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.

• 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.

• 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.

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

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

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

• 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.

• 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.

• 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.

• 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.

• 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.

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

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

• 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.

• 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.

• 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).

• 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.

• 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.

• 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.'

• 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.

• 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.

• 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 ..

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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

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

• 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.

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

• 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• 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.

• 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.

• 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 ?

• 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.

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

• 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.

• 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).

• 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).

• 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).

• 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?

• 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.

• If the two interfering waves, being still of same length X, be in opposite phases, or sÃµ 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.

• 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.

• 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.

• The moment of the resultant force R of the wrench about this line is Rr sine, and that of the couple G is G cos 9.

• It is impossible to give more than approximately accurate statistics of the resultant depopulation of Portugal; but it seems probable that the inhabitants of the kingdom decreased from about 1,800,000 or 2,000,000 in 1500 to The Slave thus discredited; the peasants sold their farms and p emigrated or flocked to the towns; and small holdings were merged into vast estates, unscientifically cultivated by slaves and comparable with the latifundia which caused so many agrarian evils during the last two centuries of the Roman republic. The decadence of agriculture partly explains the prevalence of famine at a time when Portuguese maritime commerce was most prosperous.

• If now we assume the water to have a depth d above the base, the total water pressure represented by the triangle kbh will have its centre at d/3 from the base, and by the parallelogram of forces, assuming the density of the masonry to be 2.5, we find that the centre of pressure upon the base bc is shifted from the centre of the base to a point i nearer to the outer toe c, and adopting our assumption of uniformly varying intensity of stress, the rectangular diagram of pressures will thus be distorted from the figure bfgc to the figure of equal area bjlc, having its centre o vertically under the point at which the resultant of all the forces cuts the base bc. For any lower level the same treatment may, step by step, be adopted, until the maximum intensity of pressure cl exceeds the assumed permissible maximum, or the centre of pressure reaches an assigned distance from the outer toe c, when the base must be widened until the maximum intensity of pressure or the centre of pressure, as the case may be, is brought within the prescribed limit.

• In their exposition, an historic character is first the product of his time, and his power only the resultant of various forces, and then his power is itself a force producing events.

• This condition is never observed by the universal historians, and so to explain the resultant forces they are obliged to admit, in addition to the insufficient components, another unexplained force affecting the resultant action.

• The resultant craters provide a refuge for wildlife, whilst the fallen tree trunks become a haven for beetles.

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

• State what is meant by the resultant of a number of forces.

• Find the resultant of two vectors using the vector triangle.

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

• In such circumstances the university will take all reasonable steps to minimize the resultant disruption to students ' studies.

• From the resultant free kick Liverpool won a corner.

• Justin Cochrane stroked home the resultant penalty to seal another draw at York Road.

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

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

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

• Hand harvesting and judicious use of modern vinification techniques ensures the resultant wines are packed with fruit and thoroughly modern in style.

• A clear failure to observe such proportionality will vitiate the resultant exercise of discretion whether to make either such order.

• The resultant blues are deeply funky affairs brimming with spiky guitar licks, rolling bluesy piano and Otis 's emotion-packed, world-weary vocals.

• The resultant wine was aged on its yeast lees until bottling to produce a refreshing yeasty character and rounded mouth feel.

• Most of us simply change the bulb for one with a higher power rating when we find the light inadequate for whatever reason - not realizing that it creates resultant heat and damage to the lamp and its parts.

• Following her suggestion, Charlemagne planted white wine grapes, and the resultant wines provided a colorless alternative that kept his facial hair pristine.

• Incorrect placement of the transducer may detect a pulsating maternal vessel with a resultant swooshing sound (uterine soufflé), and the rate will be the same as the maternal pulse.

• Most patients are embarrassed to admit to hair pulling and the resultant sequelae, and elicitation of this behavior is difficult.

• Treatment of the resultant medical complications of hair pulling must be addressed.

• Very likely, everyone who views the resultant pictures will have a favorite choice, as they are all very different.

• The resultant season deals with the struggles of the Ewing family to go on after losing Bobby.

• The resultant atmosphere of distrust among the public has caused many executives to see the wisdom of having positive corporate mission statements and a strong ethics policy.

• The increase in calorie burn resultant of additional muscle is fewer than 100 calories per day.

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

• 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.

• 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.

• 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.

• 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.

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

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

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

• Thus, the thermal equivalent of the unit of resultant electrochemical change in Daniell's cell is 5.66 - 3.00 =2.66 calories.

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

• The resultant being a product of mn root differences, is of degree mn in the roots, and hence is of weight mn in the coefficients of the forms; i.e.

• 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.

• 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.

• 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).

• 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.

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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

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

• They do not recognize it as a power inherent in heroes and rulers, but as the resultant of a multiplicity of variously directed forces.

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

• 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.

• 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."

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