DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
181 
For a 3 — j <p (— a 2 + £ J c 2 ) dr, this leads to 
Jo 
A’7r { — <b"'—| t"—|t—|-+£ 2 (— ir+|-)} 
+ X’71^ jf - 1 (ir* + |r + -H-) - |r —V + C - i t 2 - 4t - f) 
+ f(-«T +¥)+c(-f+ * 
+ X 7 ( /c, ) - f -( JfT T | 7 ) + 5 1 - | T- - J,? T - 5 H 2 + f ( - | T i- ) 
+ X*V{Af-* + f-‘(|T + «) + ST-f + f(|T-#) + |f» + {*(|T-*)-|f}. 
The terms in c 3 — | f ~Q> (— T £ 2 c 2 ) are similarly 
•1 0 
\%»{r S (|-T S + |T) + |T-iT 3 } 
+ 1 ■ 6 T " ' g + ( „ ) i 3 (, + ( 1 (It" + -!}t + V ) 
-¥r-|T 2 + ¥r+|(-iT + H 
+XW {f-* (-§ -*)+%-*+{-’ (tr+V-) + t-' (tr-V) 
-l9-4?T+f(-|T+V-)-ir 
+X*v (-|t+«) 
+tff+ J #T+f(2T-f)-ir}. 
It is easy to see that the terms in A 3 in a 1} c x are respectively 
ix%(r 3 -n and \%ar 5 -£+i), 
neither of which contains - Thus up to A 3 we have, in the preceding notation 
= —A h, yi — — X J k 2 , 
a 2 =^X a (h a +%k 1 a )-\ s (hk a + ik 1 k 2 ) t y-> = —A 2 (h 2 + §hk x — 2k 2 ) + yX :i hk 2 , 
Ot 3 = - A 3 /? 3 - f A 3 h% - -% S -X 3 hk 2 + ^\% 3 , y 3 = A 3 /F + ^A%%! - ^X 3 hk 2 + -^A 3 ^ 3 . 
Thus 
a i + a 2 + a i{ = -A/i + |A 2 (/? 2 + P 1 2 ) + X 3 (-/r 3 -|^ 1 -Wi 2 + ¥^ 3 -^^-pA> 
yi + y 2 + y 3 = A 2 (-h 2 -%hk 1 + 2k 2 -k 2 ) + X s {K i + ~ 7 £h%-^hk } 2 + ^-k 1 s + ihQ. 
