1921-22.] Linear Differential Systems and Integral Equations. 53 
Let us investigate the even periodic solutions of (29) which are expres- 
sible in finite form ; they must necessarily be included in 
u{x) = ®(cos Zx-\-c cos x), 
where c is a constant to be determined. We have the two equations 
A 
A 
^ ~4 ^ 3s(cos 3s + c cos s)ds 
4 /o 
qI sm 2 s g^'cog 3s-f c COS s)ds, 
which may be written 
in which 
Iw 
Hence 
=/: 
1 = gfL + L + + L)] 
+ 1-2 + ^’( ^-0 + h )] 
gZ sm' s 2m.s ds = ^ ) 
oL + f2 + dL + D 
C = o^i 
(31) 
(31a) 
L + L + + D’ 
By the use of the recurrence relation 
l^Im = ~ fw— i) 
this equation may be reduced to 
Id^ -1- 2(4 - /)c — 3? = 0, 
whence we conclude that the two even solutions of (29) expressible in 
finite form are 
4_/+{(4_Z)2_i_3/2p 
_ phi sin2 X 
and 
u^{x) - 
^(^) = e« sin^a; 
3.r + 
l 
COS 3,r -t- 
4_^_{(4_/)2 + 3^2p 
] . . (32) 
:] . . (32a) 
{Issued separately March 24, 1922.) 
