XVII.] ON HILL'S METHOD OF TREATING THE LUNAR THEORY. 79 



Let the additional terms that we seek be &x, Sy, which we shall 

 suppose so small that their squares and products may be neglected, let us 

 consider first the terms which are multiplied by the first power of one of 

 the new arbitraries, the original particular solution corresponding to the 

 case in which this arbitrary is zero. 



Then 8x, 8y are determined by the equations 



d' 2 Sx , dSy d-R ~ dR . 



-, - 2n' -jf = -T3 dx + -j-,- By + X, 

 dt at dx~ dxdy ' 



,dx d*R s ^ 



+ 2ri -j- = -js- OX + -j-,8y + Y, 



-sj -j- -js- -j-, 



dt dt dxdy dy 



where X, Y are supposed known functions of x, y or of t, and have been 

 added here to include disturbing causes not allowed for in the above form 



of R. 



, dx dy 

 Multiply the original equations by -j- , -j- and add : 



d*x dx d~y dy _ dR dx dR dy 

 df dt dt" dt dx dt dy dt 



_dR 



= ~dt ' 



since x, y are the only functions of t that R involves ; whence 



where C is an arbitrary constant ; this is the integral known as Jacobi's 

 Integral. 



Let us write 



dx d - 



then we have V = '2R+C; 



and from the original equations themselves 

 dV 



- 



? dt 



dR dR 



