BOUNDARY PROBLEMS AND DEVELOPMENTS. 71 



Section V. 



The formal solutions of the equation 



Y'(x). = [A(x)\+ B(x)}Y(x).. 



Returning to the homogeneous equation let us consider the nature of 

 the solutions when the matrix coefficient of Y • is made to depend 

 linearly upon a parameter X which is free to take on all values in the 

 finite complex X plane. To this end we shall study the equation 



(19) F'(ar). = { A(x) \+ B(x) } Y(x)-, 



where A(x) and B (x) are matrices of continuous functions, by making 

 the assumption that the equation has a formal solution 



(20) F(.r). 9 e X i y(x)dX \l\(x)- + i I\(x)- + i P,(x). + ."..{ 



where a is any chosen constant. 



If n = 1, equation (19) can be directly integrated and is seen to have 

 an actual solution of the form (20). The passage to formulas (45) can, 

 therefore, be made directly in this case, and hence we shall assume 

 in the intervening work that n ^ 2. 



X 



Setting / y(x) dx = Y(x), and substituting the form (20) in the 



a 



equation (19), we obtain the formal identity 



X7(.r> XrU) | Po(») • + \ Pi(x) • + . . . | + e"™ j Po(x) ■ + . . . | 



= { A (x)\ + B(x) ) , xr(l) P Q (x) • + i P^z) ■ + ...(, 



from which it is seen, upon equating the coefficients of X, that 



y(x)P (x)-=A(x)Po(x)-. 



This is satisfied by a vector Po(.r) • not all of whose elements vanish, 

 when and only when 



(21) \ aij (x) -S tn (x)\ =0, 



