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.
There is no difficulty in expressing the resultant by the method of symmetric functions.
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.
Temperance and the resultant health and vigour).
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.
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.
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.
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.
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."
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) ï¿½ ï¿½;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..
This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms.
+al pxp = 0, a21x1 +a22x2 + ï¿½ ï¿½ ï¿½ +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.
Forming the resultant of these equations we evidently obtain the resultant of f and 4,.
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.
Bezout's method gives the resultant in the form of a determinant of order m or n, according as m is n.
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.
= 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.
Hence this product is the required resultant of the three equations.
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.
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.
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 ï¿½; hence the resultant of and f is, disregarding numerical factors, y,y2...y,,.
(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.
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.
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.
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.
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.
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).
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.
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.
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.
For the resultant force at P, F=-VF r 2.
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.
Principal axes of the crystal, the actual magnetization will be the resultant of the three magnetizations along the axes.
5), we have for the resultant force at P R = - 2cosO X m/d' = - elm/d 3 = - M/d3.
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.
To find component forces equal to the composite or resultant force, the sum of the components must equal the resultant.
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.
There are two gravitational fields which sometimes reinforce and at other times diminish each other and the effect is always a resultant one.
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.
The sum of the suffixes in each term of the resultant is equal to mn.
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.
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 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.
They do not recognize it as a power inherent in heroes and rulers, but as the resultant of a multiplicity of variously directed forces.
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.