DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
163 
and so 
flence 
Q 0 w+t (u) . h = Q* (u ). h/e ic ' w , 0 \h~\ 
\ 0, e iC2W 
Q 0 w+t (u). h / e - iCl{w+1 \ 0 \ = Qq (u) . h / e~ ic '\ 0 
V 0, e -iot(u,+t)J \ 0j g 
This shows that the matrix on the right lias period w. Put then 
P 0 * = Q* {u) h / e ~ ici \ 0 \h~\ 
\ 0 , e~ iC2t J 
which has period tv, and is such that P 0 ,,; = P 0 ° = 1. The matrix QJ ( u) can therefore 
be written in the form 
IV (u) = P,/ . h/e iClt , 0 \h~\ 
\ 0, e tC2 y 
which is the theorem in question. 
We now compare this with the form of solution of the original differential 
equations by the method of successive approximation followed by Lagrange, 
Laplace, and others. We have 
V'V 0 \ = 1 +it/c 1> ° \ + 2 j An 0 \ + ... ; 
0, e 
icit 
Vo, w 
0, c 2 
thus 
tv (u) = Po f + tP {hyh- 1 ) + i- P {hy 2 Jr l ) + ... , 
where P is written for P u 4 , and y is written for 
V, 0 
0, ic 2 
If then, as in Laplace, ‘ Mec. Cel.,’ Liv. II., Ch. Y., t. L, of the edition of 1878. 
p. 266, we obtain the solutions of the differential equations in the form 
(P 0 4 + £A + fT>+...) x°, 
where A, B are certain periodic matrices, and x° is a row of arbitrary constants, 
we can obtain the constants ic 1 , ic 2 , which are the most important quantities in many 
applications, by taking the matrix A, which arises as the coefficient oft, and is equal 
in our notation to 
p»‘ (MM, 
