Problem of Three Bodies. 1 1 



From the last we find by means of (C), 

 xdd s — y dd% + zdd x = 0, x 1 dc 3 — y' dc%+ z' d c x = 0,1 

 xdd^-ydd'^ + zdd'^O, x'dd' 3 -y' dd'^ + z'dd'^O, L(H.) 

 x"dc 3 —y"dc^z"dc x — 0, x"dd 3 — y"dc , 2 + z"dc' 1 = O.J 



These may be found immediately from (B.). We may find 

 the corresponding equations of the first order from (E.). 

 They are 



NMm(c 3 ^'-c 2 2/' + c 1 2 , )=/t(c'V-c'y + c2')-m(c"a;-c'y + c2) > " 

 NMm'(c' 3 ^-c'^+ d A z)z=/J(d'x—dy + cz) —m' {d'x 1 — dy' + cz')> 

 NMm{c 3 x"—Cc l y" + c 1 z")=fjt,(d'x'—dy' + cz')—m(d , x—c'y + cz) 9 

 Nmm'(c" 3 .r— c" 2 y +d' 1 z)=m' (d'x 1 —dy' +cz') +m(c" x—dy+cz), 



and two others. The second members, where c", d, and c are 

 constant, manifest how these equations are formed. 



We might derive from (B.) many other curious results, 

 but we shall only notice the following : — 



c 



Mm^ + Mm'^+mm'^j± = 0, 

 ^ + Mm'^+mm'^ = 0, 

 M m —^ + Mm' —~ + mm' —^ = 0. 



M»» 



(K.) 



However convenient the perturbating function of Lagrange 

 may be for the purposes to which it is applied, it is not adapted 

 to the finding of integral equations of the first order. 



Make R = 



Mm Mm' mm' 



+ 



+ 



may be put under the following form : — 

 d?x dR dR 



and the equations (A.) 



Mm 



dt* 

 d* 



= ^ +m ^" 



Mm d¥ 



d 2 y dR dR 

 dy dy' 



dR 



Mm-— = //,—=— 



dt* 



dR 



a lu +m lu''> 



Mm 



= fi' 



M m' — A — u! " + rn'^< M m 



df- 



dR 



''"'dy' 



dR 

 dy 9 



,d*x' _ ,dR 



dt 2 ' 



, d*z' 



dt 2 



+ ml 



dR 



dx 3 



d x 



,dR ,dR 

 fi' -7-j + m' -j— . 

 az' dz 



Wl. 



These may be put under the following form : — 



Mm 



d*x 



+ mm 



d?x 



-d*x' _ „dR 

 dt* = dx* 



,, .d^x 1 .d^x-d^x 1 

 M m -r-o — mm' 



dt* 



dt* 



„dR 



