DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
145 
similarly, draw a perpendicular MK from M to SY; denote SK, KM by A + £ and >?. 
If R = SE, r — EM, p — MS, we have, with proper conventions of sign, 
lt“ = (A + xY + y 2 , p — (A + ^) 2 + /, 
-r" = [i(A + g)—^*i \/3 —A — t x]"+ [|-(A-i-^) \/3 + %v — y]~- 
The accelerations of E, relatively to S, parallel to SX and parallel to HE, are, 
respectively, 
— (E + S) | ^ + y/3 —(A + .x) ^j- j~(A + ^)—g-^y/3 ? 
R 3 v" p ° 
-(E + S)^+M 
nA + f)\/3+^>j —y + + y/3 . 
the accelerations of M, relatively to S, parallel to SY and parallel to KM, are, 
respectively, 
-(M + S)' 
^4ii_gjT 2 (X + ^) — 2>?y/3 — (A + a;) , y/3 ^ ( A + f) -y/3 + -g-?? — y 
-1*3 
_-r-i I (A+x)+2 y V 3 
J R 3 
A —T? /i i(A + ^)+|->; —y -y/3 \ (A + f)—y/3—A—a; 
V V \ 2 ’ ?’ 3 2 ' r 3 
_-]? 3 h/~~+ (A + a:) y/3 
^ • R 3 
If, then, in the equations of motion relatively to S, after expanding in powers of 
x, y, f, r], we equate the finite and the small parts, the squares of x, y, g, tj being 
neglected, we obtain 
A-A(9 2 = --f 2 
A 2 
and 
A 2 0 = constant, = h , say, 
where 
together with 
jm = S + E + M, * 6 = % 0 = ^, & c. 
dt 
X—20Y—<9Y— (@ 2 - ^ X = 
\ A 3 / 4 A 3 
Y + 20 X + ex - l e 2 - ^) Y = 00 [(E—M)X+ v / 3 (E + M) Y], 
c& 2 
"4S + E + M 
a/3 
X+(E—M) Y 
x 
VOL. CCXVI.-A. 
