280 



where 



H= T -~ K, 



T representing llje kinetic energy and K tlie force-fnnction. In the 



problem of planetary motion we have 



k 



K = - +5, 



r 



where S is the perturbative function. According to a theorem dis- 

 covered by Jacobi, any new system of canonical variables pi , qi can 

 be derived from an arbitrary function ^P {-Vi , qi) of .r, and q;, by 

 putting 



^— = 3// ' ^— = P/ (2) 



If then, by means of (2), we replace av and /// in H by pi andqi, 

 the equations for the new variables are 



dt dqi dt dpi 



Jacobi's method of integration, which has led to the system of 



canonical elements introduced into asti'onomical practice by Delaunay, 



consists in so choosing <2» that the equations (3) are of a much simpler 



form than (I). For this purpose Jacobi chooses for 'I* an integral 



of the partial differential equation, which bears his name, and 



wdiich is constructed as follows. In the function H{.Vi,y:) replace 



d<P 

 Vi bv ^; — , then Jacobi's equation is 



O.Vi 



;'" d.vi 



The constant h is the energy of the motion. 



Tf we take /S=0, and, instead of x; , y; introduce polar coordinates 



di' ds 



r, s, IÜ, and the corresponding momenta r' = m — , s'=i mf^ — , 



dt dt 



dw 

 lo' =1 mr^ cos" s — , the energy function becomes 

 dt 



1 / s'^ iv" \ k 



B, = --(r-i-- + -^-^] = h .... (4) 



2i7n \ r r cos' sj r 



Then Jacobi's equation admits the integral 



H{ xi,^\ = h. 



Ci / 0' Ci / '^^^ G^ 



*" = 0«- + J l/e- " ^, * + J [X-'"" + ~ - 7^ *• 



where Q and G are constants of integration. Jacobi now takes 



