132 
PROF. H. F. BAKER ON CERTAIN LINEAR 
which, as v/u is not 2 or —2, determine A 2 , A_ 2 , and C 2 , the last being expressible by 
means of the given coefficients of u, v, u 1} v 1} iv v 
Proceeding similarly with the general value of n, we at once reach the conclusion 
stated, the absolute term in w n being determined in terms of the coefficients in 
U > U l> U n- 1> A As • • ■ 5 Ai-15 W lt u, 2> • • • 5 W n -V 
§ 2. Now consider an equation 
A^+2pA+Cx 
(It (It 
o, 
where, w T ith f = e T , A, B, C have the forms 
A = « 0 + A (cq£+a_X _1 ) + A" (w 2 (,“ + a_2<, " + <%>) + ..., 
b = & 0 +A(^+6_ 1 r i )+x 2 (6/ 2 +6: 2 r 2 +u+... 5 
C = c 0 + A (cj^+ c_N _1 ) + A 2 ( c 2 £ 2 + c_ 2 £ _2 + c 20 ) + • • • j 
which are periodic functions of r, with period ri, capable of being arranged as 
power series in a parameter A, the coefficient of A r being a linear function of 
r, r- 2 ,..., r-b r r . 
In accordance with the well-known theory of linear differential equations with 
periodic coefficients, we substitute 
x = e KT X, 
where k is a constant, and so obtain a differential equation 
AX"+ 2 (atA + B) X'+ (A/c 2 + 2B/c+C) X - 0, 
which, when k is properly chosen, is to be satisfied by a periodic function X. That 
this is possible follows at once from § 1, as we now explain. 
First we can draw some inference as to the form of k. For compare the original 
differential equation in x with the equation obtained from it by changing the sign of 
A in each of the series A, B, C. It is clear that the new differential equation may 
equally be obtained from the original equation by change of t into r + 7r?’, which 
changes f into — g; this latter change, however, only multiplies the factor e KT by the 
constant e ,7r< ; the factors e KT appropriate to the two independent solutions of the new 
differential equation are thus the same, in their aggregate, as the factors for the 
original equation. Thus the change of the sign of A changes the two factors e KT 
appropriate to the two independent solutions of the original differential equation 
among themselves, either by leaving both unaltered or by interchanging them. 
Assuming that k is capable of expression as a power series in A, 
k = «r 0 + /qA + /c 2 A 2 + ..., 
