72 BIRKHOFF AND LANGER. 



i.e. when y(x) satisfies equation (21). For any given x, say x = a- , 

 however, the left-hand side of (21) is a polynomial of degree n in 7(0-0). 

 Hence the equation is satisfied by n roots 7i(.To), 72(.fo),- • -7n(^o). 

 We shall assume that for the case in hand these roots can be grouped 

 into the ?* functions 7i(.r), 72(2),- • -7n(-i'), which satisfy the three 

 conditions 



!(i) ji(x) continuous, i = 1, 2,. . .n, a ^ x ^ b, 

 (ii) yfa) ±vi(x) for j±i, 

 (hi) 7<(*)=t:0, *=l,2,...n. 



Clearly the last condition can be fulfilled only if | A | 4= 0; we shall 

 assume this to be the case. 



Consider now a change of the dependent variable in equation (19). 

 Setting Y (x) ■ = <J>(a-) Y (x) • , where 4>(.r) is a matrix whose elements are 

 continuous as well as their first derivatives, a ^ x ^ b, and | <£> | d£. 0, 

 the equation becomes *T- + $T-' = [A\ + B) 3>F-, that is 

 y'. = {$-1 ^l$X + $- ] B<S> - 4>- ] $'} y- It is evident here that if the 

 coefficient of X is not originally / then no such change of variable has 

 the effect of reducing it to I. In the subsequent work it is desirable 

 to transform A(x) into the matrix R(x) given by R(x) = (5,-,-7,(a0). 

 By such a transformation the formal solutions (20) are carried into 

 others with the same functions ji(x) in the exponents. 



Since the <b(x) in question must satisfy <£ _1 A$ = R, 7 it must fulfill 

 the conditions 



(i) AQ = QR, 



(ii) I Wx) I =*= 0. 



To satisfy condition (i) the elements of 4> must be solutions of the 

 algebraic system 



n 



S a ik <p kj = (fij 7j, i, j, = 1, 2, . . . n, 



while the possibility of fulfilling this condition by means of a matrix 

 $(.r), none of whose columns contain only vanishing elements, depends 

 upon the existence of a solution for each of the n linear systems 



7 For fuller discussion see, for instance, Bocher, loc. cit. Chap. XXI. 



