82 BIRKHOFF AND LANGER. 



It is apparent from this that Six) is a solution of a homogeneous 

 differential matrix equation of the type 



(39) S'(x) = J \R + B 4- ^ $(.r, X) j S(«). 



The elements of $(a-, X) are power series in ( - ) with coefficients 



which are continuous in x, and are seen to be convergent, since these 

 elements are rational in X. 

 But equation (26) can be written in the form 



(40) Y'(x)= pl(.x)\ + B(x) +^*(x,A) j Y(x) -j-jjjSfeX) Y(x), 



and considering this as a non-homogeneous equation we know, in 

 virtue of the developments of page 62, that its solutions, i.e. the solu- 

 tions of (26), are given by 



Y(x) = S(x) C + h(x) T(t) | - ± Ht, X) 7(0 | dt, 

 i.e. by 



(41) Y (x) = Six) C- -1 C S(x) fit) Ht, X) 7(0 dt, 



where Tix) = S~*(x) and where the lower limit of integration, which 

 has been omitted, may be chosen at pleasure for each of as many parts 

 of the integrand as desired. 



Substituting for 7(.r) in equation (41) its equivalent as given by 

 the form 



(42) Y(.r) = Uix) S(x), 

 we have further 



