DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
J 77 
To find 
a 3 = </> (— a, + £c 2 ) cZt, c 3 = (—«.,+ £c 2 ) dr 
Jo Jo 
up to A 3 it is sufficient to take <p = A [_h + h l (£ _1 + £)] 5 so we obtain 
<p (—a 2 + £c 2 ) 
= A 3 {h s {-^T 2 - 2r-3-rf+3f) 
+ h%{-±T 2 t 1 - M" 1 -2r i + i-M-W“4f-T£ a +ir) 
+m x 2 (M^+^^+Tr^K^+ir+ST+i+Tf-K+rr-K 3 ) 
+V(-M- 8 +M- 1 +Tr i +K- 1 -t+M+K+Vf-tf a -W 8 +? a ). 
and hence 
Oa 
j (p ( — a 2 + fr 2 ) dr 
Jo 
=• A 3 A 3 ( — -g-r 3 r 2 3 t 4 + 4£) 
+ \ 3 h% [(iT 2 + |T + |)r i + |- T -f+(-lr 2 + iT-|)^(-iT + f)f] 
+ A 3 /J^ 2 
(-It-*) r 2 +(-r-f) rH^ + f^ + ir + ^+^-V-) t 
+ if-i)t 2 -K 3 
+ ^ 3 [«- 3 + (-iT 2 -2r-V-) r i -fT + |- + (|r 2 -lr + ^) {-%? + (-& + &) ?\ 
Similarly, 
Co = 
I f V J (-®a+foj) dr 
Jo 
= \W[(jT ! +3T+6)i'-' + (-i-T ! + 3T-6)] 
+ X« 1 [(ir’ + T + f) 0 S -K- 1 + (-iT 3 -iT 2 -4T-f) + (-r + i)g 
+ X 3 A^[(-Jt-M) f- 3 -i( T+ l) f- 3 + (-J T 3 -4 T -¥) f-‘ 
. +iT 2 -§T + W-+Ti-4-K 2 ] 
+ X*i,*[*r + (-^-|r-|»)f-'+|{->+*V+iT'+VT+*-|f+(-jT+*) n 
Picking out now the terms in r, putting therein £ = 1, and using the notation 
previously explained, we have, up to A 3 , 
15 
a. 1 = —A h, y 1 = —A k 
a 2 = A 2 (h a —2hJc l —jfki) +A 3 (PA), y 2 = A 2 ( — 2h 2 + hJc 1 ) +A 3 {ihh 2 -l\k,) t 
a, 3 = A 3 (- 4/r 3 4- 6h% + 2M 1 2 - 1 /T 1 3 ), y 3 = A 3 (6A 3 -4/i^ 1 ~- 1 3 9 -M 1 2 +1^i 3 )- 
VOL. CCXVI.-A. 
2 B 
