BOUNDARY PROBLEMS AND DEVELOPMENTS. 79 



(34) Z'(x) = -Z(x) [R(x)\+ B(.r)} 

 be obtained. This T(x) has the form 



(35) T(x) = (cie-™iW-BiM[8 i3 ]X 

 and satisfies an equation of the form 



T'(x) = - T{x) \\R(x) + B(x) +~M(x, X) |. 



Moreover each row is a formal solution of the vector equation 



(36) >Z'(x) = - -Z(x) {R(x) X + B(x)}. 



Considering differentiation as merely a formal process defined by 

 the usual rules it is readily seen that the differentiation of the formal 

 matrices S(x) and T(x) is permissible. Hence we have 



d 



— TS = TS' + T'S 

 ax 



= TUR + B+\l\s- T\\R + B+- k Mls 



= ~T{L-M}S. 



Since 



IS = I i ae WihlkCje WhjlkJ 



we find upon differentiating, and removing the exponential factor, 



( ( a (1) (T {2k) ) ) 



CiC,- J {X( 7 ;-7i) + bjj-biil |« l7 + -^- + . . . + ^£- | j 



