DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
131 
wherein 0 r is a linear function of £ r , £ r ~ 2 , £ r ~ 4 , £ 4-r , £ 2 ~ r , £~ r , and this is a periodic 
solution. Its period is 2 iri ; we can, however, if we wish, express the same result 
with a period 2?r by writing t = it. 
For the substitution of the assumed form for X requires the identity 
(u + 2AX) 2\V ;, n + (v + 2AX) 2\y B + 2A n «y (1 + 2A n 0„) = 0, 
which, equating the coefficient of A” to zero, will be true if 
w<p"n + u \<k> n n -\ + • • • + Un-lty" 1 
+ V <p' n + V l<p'n-1 + • • • + V n _- l (p\ 
+ W l<Pn-\ + W 2 ( p n -2 + ••• + W n-l ( Pl+ W n = 0 - 
In particular for n = 1 
U(p'\ + V({/ 1 + iv 1 = 0. 
If herein we suppose 
0i = Ajb +A_j(, , w 1 = cX+c_X , 
u, v, c 1} c_! being given constants, we obtain 
^ (A^+A_X -1 )+v (A x ^—A_ 1 ^ _1 )+cX+c_X T ' 1 — 0; 
which is satisfied by 
A x = 
Cl 
u+v’ 
A_! 
C-l 
-w —v ’ 
For n = 2 the condition is 
u<f> f 2 -\-v<p r % + ?q0\ + i\<p\ + = 0. 
"Writings 
Ui = «if+a_X _1 5 iq = b^+b_^-\ u\ = G^+c_^-\ w 2 = c 2 £ 2 + c _ 2 £- 2 + C 2 , 
and assuming a form 
02 = A 2 ^ 2 + A_2^ _2 , 
the condition becomes 
4 u (AA 2 +A_ 2 r 2 ) + 2v (A 2 f 2 -A_X- 2 ) 
+ (cqf+a&_if _1 ) (AX+A_X _1 ) + (&if+&_if“ 1 ) (AX—A_X -1 ) 
+ ( c i£+ c -i£ *) (AX+A_X _1 )+c 2 <y + c_X “ + C 2 = 0 ; 
equating the coefficients of £ 2 , f -2 , £° to zero, we obtain 
(4:U+2v) A 2 = — rtjAj — ZqAi—c^—c 2 , 
(4w— 2v) A_ 2 = —a_iA_ 1 + 6_ 1 A_ 1 —c_ 1 A_ 1 —c_ 2 , 
C 2 = —WiA.i—a.iAi + ^iA.! —h.iAi—CjA,!—c.jAi, 
t 2 
