Problem of Three Bodies. 9 



Making x, y, z, and m change places with #', y' f z' and ml ; 



d 2 x' dQ 



+ 



= 0, 



" v + 4^ = o, 



**■+*£*• 



<// 2 ' da/ ~ "' </* 2 T iy* ~ "* d* 2 ' «?*' 

 for the motion of m. And similarly for the motion of ml about m ; 

 r/ 2 *" dQ" _ d 2 y" dQ' « d 2 *" rfQ" 

 T -0 ' "dF + 



+ 



17 = 0' 



+ 



0. (A.) 



dP n rf*" -v ' dP ' rfy ~ "' rf* 2 ' tf*" 

 By substituting for Q, Q', and Q" their values, we shall 



easily find 



xd 2 y—yd 2 x . . ,, { m' m'\ 



d/ ' = {xy ~ X7j) ■ \w " vph 



ad* z — zd 2 x _ , ,. (ml ?n'\ 



jp -{XZ xz).\jps -pjg J, 



yd 2 z — zd 2 y , , „ / w' w' \ 



a* d 2 y 1 — 1/ d 2 x' , , .. ( m m\ 



-*ir — = { *y - xy) - \7* - -ph 



x 1 d?z' — z'd^x 1 _ .. f . / m m\ 



— \Jf Z — X Z ) . ^^3 — -^J) 



df 



(B.) 



?/ d 2 z' — d d? y' , . .. ( m m\ 



~ — J? — - = (y'*-y*') • {jm -75-j; 



x"d 2 y» -y"d 2 x" , , I /M M\ 



, , /M M\ 



a"^'-*"^*" 



Uc.) 



</2 2 

 !/'*z!'-2»d* ! /' _ /M M\ 



<^ 2 . -^~-^^vpr~w-_ 



From the last we derive without difficulty, 



a? d 2 y — yd 2 x_x' d 2 y' — y' d 2 x' _ a?" d 2 y" — y ri 2 x" 

 x~d 2 z- zd 2 x " x'd 2 z>-z'd 2 x ~ x" d 2 z" -z" d 2 x"' 

 x d 2 y -yd 2 x _ a/ d 2 y> -y'd 2 x' _ x" d 2 y" -y" d 2 x" 

 y d 2 z - z d 2 y ~ y d 2 z' - z' d 2 y' ~ y" d 2 z" — z" d 2 y"' 

 From these two, dividing the one by the other, we obtain a 

 third set. 



From the same source also we find 

 (x 1 z — z 1 x) (x d 2 y —yd 2 x) = (x 1 y — xy') (xd 2 z — zd 2 x), 

 (x'z-z' x) (V d 2 y' -y' d 2 x 1 ) = {a? y -y 1 w) (x 1 d^z 1 - z' d 2 x% 

 (x'z - zlx) (x"d*y"-y"d*x") = (x'y -y'x) {x"d*z"-z"d*x"), 

 (y'z — z'y) (x cPy —y d* x) = {rfy—y 1 x) (yd 2 z — z d 2 y), 

 and several others. 



(D.) 



