DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
1 61 
In this last equation, put 
leading to 
then we have 
Writing 
these are 
or, say, 
_ 1 n int I dtC 
X = ie 
dt 
—mx , 
ini 
— inx = Xe _ “ ( — Ye 1, 
dX _ i\Js 
Y = ^e~ int +inx j , 
dt = Xe~ int +Ye int ; 
dt 
- -Y e int (Xe~ int -Ye int ), 
2 Yb 
~ = - ^ e~ int (Xe~ int —Ye inl ). 
Cuts 2 71 
2 it = T, £ = e T , 
^ = - ± (X-YC), - ± (I(--Y), 
dr 4 n y dr An v 
\r% -i 
where, as is usual, the single quantity — written before the matrix, is to be 
multiplied into every element of the matrix. 
In particular, when n = 1, \\r = Aa cos ht + Ab cos kt, 
L(x,Y) = («p+M(x,Y),. 
where p, denote the matrices 
P = i(f 4A +f -4 *)/ -l, f 
Ua 1. 
Thus the solution is expressed by 
(X, Y) — O (ap+bq) (X°, Y"), 
where Q (op + bq) is of the form 
1 + aQ p + bQq + a 2 QpQ,p + ab {QpQq + QqQp) + YQqQq + ..., 
and we have proved that this series is uniformly and absolutely convergent. 
If we assume such a form of solution it is easy by successive steps to obtain the 
values of the coefficients independently of the method we have adopted. What is of 
present importance is that we have shown the series to be convergent, a fact which 
appears to be denied by Poincare, 
vol. eexvi.—A. 
z 
