A 
DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 171 
and vanishes when p 1 = p^ — 0, p 2 = — 2 np 3 . But the roots of the determinantal 
equation 
| /3 -1 a — \Js | = 0 
are all different, and therefore the invariant factors are linear. 
The roots are 
yjr = ± ^{(28) 4 +lp, i/r = ± m{(28) 4 -lp, 
of which only two are pure imaginaries; thus not every disturbed orbit is periodic. 
§ 18. We pass on now to give the details of the application of the general method 
above explained to the computation of some particular cases. 
A very simple case may be first given, merely as an example of the notation and 
method, since the results, once obtained, are easily verified. 
Take the equations 
dx 
These may be written 
d (x, y) = 
dt 
or, say, 
where 
We have at once 
2 = —x cos t + y (l + sin t), 
2 r ly = — x (l — sin t) + y cos t. 
■iA/L/ 
Jr / 0, 1 \ +jf / — cos t, sin t' 
\ —1, 0/ \ sin t, cos t y 
d ^ = {u + v) (x, y), 
(%, y), 
u 
_ i 
2 ( °’ 1 
V — 1 , 0 
(2m) 2 = / 0, 1 
U 0 
v = w / — cos t, sin t ' 
sin t, cos t, 
0 , 1 \ = - 1 , 
-1, o 
Q(u) = l+wi + ^|i 2 + |q^+..., 
= 1 -2p« !+ n^ 4 -- +2 “p-^ ( ^ 
= cos \t + / 0, 1 \ sin \t, 
— / cos -%t, 
L, 0, 
sin \t 
— sin cos \t, 
2 A 2 
and therefore 
