PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
525 
dx^ + dy"^ = dr^ + r^df, 
(rfR) c?R dr ^ 
dt d^ dt' dr dt ’ 
and the equation (4.) becomes 
Again, the equations (1.) and (2.) give 
d'^ii d^x <?R c?R 
(5.) 
But 
d.r^^— 
d^x ' dt 
dy‘-y d^x dt , c?R «?R «?R 
^ip-yi?=-dr’ ^Sj-y-S:=i»- 
Hence, by integrating, 
7 r«?R , . , 
(®-> 
h being an arbitrary constant. Consequently, by substituting for ^ in (5.) from (6.), 
and neglecting the square of the disturbing force, 
M M , MCdU , Cm/Cd'R. , \ , , <i« A , 
}M-dl‘‘^=dt]Ti'^^-]ir\]T>‘^*y*’ and ^=^5 nearly. 
Hence it will be seen that to the first power of the disturbing force we have 
df-r^i— ^ +C— 2J dtj — . 
( 7 .) 
The equations (6.) and (7.) are suitable for determining the forms of the develop- 
ments of r and & in terms of the time. 
The function R becomes, by neglecting m!z, 
m\xod + yy^) 
m 
and reckoning & from the axis of x on the plane of the orbit of m! in its position at 
the time TJ,, we have to the same approximation, 
a?'=r'cos^', 3/'=r'sin ^'cosiy. 
a'=r'sin & siniw. 
Hence 
77i / 
R =-72 (cos ^ cos ^' + sin & sin 4' cos a)- 
m 
(p 2 j ,2 _ 2rd (cos fl cos 4 -f- sin 6 sin fl'cos cu) 
If powers of a above the second be neglected, the following approximate value of R 
is obtained ; 
R=^ cos(^— 4)' 
m' 
(,d2 _|_ ,.2 _ 2f^J cos (9 — 4 ) ) ^ 
00 / 7 ' 
■2m' sin 6 sin 4sin^7; 
2\r 
,J-2 
(r'^ _j- y .2 _ 2rr' cos (fi — 4) ) 
4 
3 z 2 
