362 Royal Society. 



represent any two of the 2w constants a^ &c., J, &c., then [^, q] is 



either =1 or =0, according asj9, q are & conjugate Y>a.\T «;, 6,, or not. 



Next it is shown that if «,, ffo, ...«„ represent any functions of Inn 



variables a; 1 ... a;„, y, ...y„, satisfying identically the -^ ^ con- 

 ditions [«,, a^]=:0, then if by means of the n given equations ex- 

 pressing «), &c. in terms of the variables, the set y^, y^ ... y^'bQ 



expressed as functions of x^ ... x^, a, ...a„, the — ^ relations 



-i-i. = -^ will be identically satisfied ; in other words, the expres- 

 daSj dxj 



sion for y, ... y„ will be the partial differential coefficients of a func- 

 tion of x\ ... x„. 



Hence it easily follows, that if any n integrals «i...a„ofthe 

 system (I.) be given, which satisfy the conditions [a,-, a/\=:0, a 

 " Principal Function " X can always be found, from which the re- 

 maining integrals of the system may be deduced by means of the 

 second set of equations (II.). 



The relation in which these investigations stand to the discovery 

 of Sir W. R. Hamilton (as improved and completed by Jacobi) is 

 pointed out. And it is shown that the system of n differential equa- 

 tions of the second order 



/dW\'_dW 

 ^ dx'i '^ dXi 

 (to which Lagrange had reduced the dynamical equations, and which 

 Sir W. Hamilton had transformed into the system (I.) by a process 

 depending upon the circumstance that, in dynamical problems, W 

 contains x\, x\, ... x\ only in the form of a homogeneous function) 

 may, by means of the theorems established at the beginning of the 

 paper, be reduced to the form (I.) without assuming anything as to 

 the form of W, which may be any function Vvhatever of a.-, ...a?„, 

 x\ ... x\, and /. 



The 2m integrals of the system (I.), obtained in the way above 

 explained, being shown to satisfy the conditions 



[a„ 6J = 1, [«,, a-\ = [«,., bj-] = [b,. b,-] =0. 

 it is proposed to call them " normal integrals," and the constants 

 «, &c., $1 &c. "normal elements," any pair «;, bj being called con- 

 jugate. 



In the second section, the author gives a simplified demonstration 

 of Poisson's theorem (extended to the general system (I.)), that if 

 /, g be any two integrals, [/, g] is constant. The preceding prin- 

 ciples are then exemplified by application to the problems of the 

 motion of a material point under the action of a central force, and 

 the rotation of a solid body about a fixed point. 



In each case three integrals, c,, c^, c.,, are taken, satisfying the 

 three conditions [c„, c,,]=0, [c,, C|]=0, \_c^, c^'] = 0; the first being 

 the integral of vis viva, and the other two being derived from the 

 integrals expressing the conservation of areas. In the former pro- 

 blem the " principal function " is then found with great ease, and the 



