250 Variations of the Arbitrary Constants in Elliptic Motion. 



M m'{xx' -\-yy' -{-zz'^ 

 integral relative to x, y, z, we have Q= — — ,-^ -f- 



~^+c, c being a function of x', y', z', which may be neglected or it 



may be considered as included in Q ; it is evident that the partial 



c/Q dQ, dQ 



differential coefficients — j~' — j^' — ~7~' equal the forces in the 



d^'x dQ d^y 

 directions of a?, y, z, hence. (2) will be changed to -j~=--j-i -37^= 



dQ, d-'z dQ 



-I-' ^72~^~' (3)3 which are the fundamental equations used by 



La Place in his theory of the moon. 



Again, multiply the forces which depend on m', and which act in 



the directions of x, y, z by dx, dy, dz severally, and put the sum of 



the products =dK, then take the integral relative to x, y, z, and we 



m'(^xx^-\-yy'-{-zz') m! 

 have R= ji —~? (4), whose partial differential co- 



^ . , , ^ d^-x lAx d^ 



efficients taken as before, will change (2) to ^— -{--~ +^ =0, 



d^-y Mt/ f/R d-'z lAz dK 



^^7+^+"^=0, ^ + V"3 +;^=03 (5), which are the equa- 

 tions used by La Place in his theory of the planets. It may be ob- 

 served, that if M' and m should be attracted by another body whose 

 mass is m" , revolving about M' in the same direction as m, and 

 whose distances from M', m are denoted by r'" , r"", and rectangu- 

 lar coordinates x", y", z", its effect would be to add the terras 



m"{xx" -{-yy" -\-zz") m" 



7775 — ~jjy, to the terms in the value of R, and so 



on for any number of bodies whatever ; and it may be remarked 



dR dR dR 



that R is called the disturbing function, and -j-i -7-' "j~ the dis- 



° ' dx dy dz 



turbing forces. 



ydR xdR dc zdR xdR dc' zdR ydR dc" 



dx ~ dy ~dt' dx ~ dz ~dt ' dy ~ dz ~ dt' ^ ^' 



multiply the first and second of (5) by —y, x severally, add the 



J xd-y-yd'x , zd^x — xd^z 

 prpducts and we get ^-^ = dc, similarly j- = 



, yd-z — zd^y . xdy — ydx 



— dc, ^^ =dc" whose integrals are — '—^, — =c. 



