DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
173 
19. We now consider cases of the equations 
d 
dT (X,Y)=-±/l, —C\(X,Y); 
“I 
these are derivable from the equation 
by taking 
leading to 
^~+(n 2 +^)x = 0 
X = i£ e ' nt ~~inx) , Y = e~ mt + inx ), t — 2 it, £ 
x = — (Xe-* ,t -Ye , '" < ). 
n 
= e, 
As we wish particularly to illustrate the method of obtaining the characteristic 
exponents from the present point of view, we take first a case in which explicit terms 
in t arise early in the method of successive approximation. We take namely n — 1, 
and suppose 
— = A h + 2Xk 1 cos 2 1 + 2X 2 Jc 2 cos 4£+..., 
4 
= Xh + Xk 1 w 1 + X 2 k 2 w 2 + ..., 
where A is small, and w r is used to denote £ r + £ r . 
Denoting by-<£, our differential equations are 
i&l) = u( x,Y ) , 
ClT 
where 
u — / -(/>, 
\-tf~ 1 , 0 / 
The coefficients in these equations have period 2-n-i ; by what we have previously 
shown (§§ 14, 15), the solution is of the form 
(X, Y) = Ph /e~ qr , 0 Xfr 1 (X°, Y°), 
V 0, e qT ) 
where P is a matrix whose elejnents have the period 2iri, h is a matrix of constants, 
and q is the constant which we particularly desire to find. As 
x — i(Xe~ lt — Ye 1 *), 
this corresponds to characteristic factors Q* l{1+2q)t for the original equation in t, whose 
coefficients have period ?r. The quantity q is to be found by determining the terms in 
