ON THE DIFFERENTIAL EQUATIONS WHICH OCCUR IN DYNAMICAL PROBLEMS. By ARTHUR CAYLEY. JACOBI, in a very elaborate memoir, "Theoria novi multiplicatoris systemati æquationum differentialium vulgarium applicandi," has demonstrated a remarkable property of an extensive class of differential equations, namely, that when all the integrals of the system except a single one are known, the remaining integral can always be determined by a quadrature. Included in the class in question are, as Jacobi proceeds to shew, the differential equations corresponding to any dynamical problem in which neither the forces nor the equations of condition involve the velocities; i.e. in all ordinary dynamical problems when all the integrals but one are known the remaining integral can be determined by quadratures. In the case where the forces and equations of condition are likewise independent of the time, it is immediately seen that the system may be transformed into a system in which the number of equations is less by unity than in the original one, and which does not involve the time, which may afterwards be determined by a quadrature, and Jacobi's theorem applying to this new system, he arrives at the proposition "In any dynamical problem where the forces and equations of condition contain only the coordinates of the different points of the system, when all the integrals but two are determined, the remaining integrals may be found by quadratures only. In the following paper, which contains the demonstrations of these propositions, the analysis employed by Jacobi has been considerably varied in the details, but the leading features of it are preserved. § 1. Let the variables x, y, z....&c. be connected with the variables u, v, w....by the same number of equations, so that the variables of each set may be considered as functions of those of the other set. And assume If from the functions which equated to zero express the relations between the two sets of variables we form two * Crelle, tom. XXVII. p. 199, and tom. xxix. pp. 213 and 333. Compare also the memoir in Liouville, tom. x. p. 337. For, representing the velocities by x', y'. .the dynamical system takes the form dt: dx dy.. : dx dy'. =1: x' y'..: X: Y.. and the system in question is simply dx dy..: dx dy'.. : =xy. X : Y.. determinants, the former with the differential coefficients of these functions with respect to u, v,....and the latter with the differential coefficients of the same functions with respect to x, y,......the quotient with its sign changed obtained by dividing the first of these determinants by the second is, as is well known, the value of the function v. Putting for shortness is the reciprocal of the determinant formed with A, B... A', B'..... &c. Or it is the determinant formed with a, B,....a', B'... ., &c. From the first of these forms, i. e. considering v as a function of A, B.. where the quantities a, B,..a', B',. .and A, B,. .A', B',. . may be interchanged provided - be substituted for v. (Demonstrations of these formulæ or of some equivalent to them will be found in Jacobi's memoir "De determinantibus functionalibus," Crelle, t. XXII). Hence — d▼ + ad▲ + ßdB......... + a'd'A' + ß'dB... ... = 0. whence separating the differentials and replacing A, A', ....... B, B',.... by their values § 2. Let X, Y.. be any functions of the variables x, y.. and assume U, V,. . being expressed in terms of u, v,. . + du dv + dX dv dv dx + + Then du dy do dy do dy + Y dU dv i.e. v. (where, for greater clearness, an additional letter z has been introduced). From these we deduce the equivalent system Suppose that u and v continue to represent arbitrary functions of x, y, z, but that the remaining function w,...is such as to satisfy W=0,...(so that w,...may be considered as the constants introduced by obtaining all the integrals but one of the system of differential equations in x, y, z...), we have Also the only one of the transformed equations which remains to be integrated is or Vdu Udv = 0, (in which it is supposed that U and V are expressed by means of the other integrals in terms of u and v). Suppose M can be so determined that (M is what Jacobi terms the multiplier of the proposed system of differential equations). Then or My is the multiplier of Vdu - Udv = 0, so that fMv (Vdu - Udv) = const. = Hence the theorem :-"Given a multiplier of the system of equations dx dy: dz.... X: Y: Z....(the meaning of the term being defined as above), then if all the integrals but one of this system are known, the remaining integral depends upon a quadrature. Jacobi proceeds to discuss a variety of different systems of equations in which it is possible to determine the multiplier M. Among the most important of these may be considered the system corresponding to the general problem of Dynamics, which may be discussed under three different forms. §4. Lagrange's first form.* Let the whole series of coordinates, each of them multiplied by the square roots of the corresponding masses, be represented by x, y....and in the same way the whole series of forces, each of them multiplied by the square roots of the corresponding masses, by P, Q....; then the equations of motion are dx dt2 = X, day dt2 = Y.. * I have slightly modified the form so as to avoid the introduction of the masses, and to allow x for instance to stand for any one of the coordinates of any of the points, instead of a coordinate parallel to a particular axis. = where 0, Þ = 0. . . . ..are the equations of condition connecting the variables, and λ, u......coefficients to be d2x determined by substituting the values of &c. in the equations d20 dt2 0, = 0 &c. It is supposed that as well dt2 Q. .as O, Þ....are independent of the velocities. In order to reduce these to an analogous form to that previously employed, we have only to write Supposing that M is independent of x', y', z',......the equation on which it depends becomes immediately To reduce this we must first determine the values of X, μ..,. and for this we have |