BOUNDARY PROBLEMS AND DEVELOPMENTS. 77 



i.e. vt^hi-y^+pf' = ?hkP%, 



whence it is seen, upon setting i = j that 



n 



„>)' — y h r>> 

 Pjj UjkPkj > 



/fc=l 



namely that pyy can be determined by means of a quadrature if the 

 quantities on the right are known. 



The determination of the elements of pjy by means of these formulas 

 depends, therefore, only upon a knowledge of the elements of P M _i 

 and upon the existence of P M _i . Moreover, it is seen that in general 

 P M possesses one less derivative than P M _i. Inasmuch as Pi has 

 already been determined, and was seen to possess k derivatives it is 

 clear that the matrices P^ for fj. = 0, 1,. . ., (k + 1), may be suc- 

 cessively determined, and that in general the matrix Pa- + i is merely 

 continuous. 



If k is finite, we can, therefore, determine a differentiable matrix 

 S(x), given by 



(31) S(x) B j Po(x) + * Pi(x) + - . . + jf P k (x) | E(x) 



which will satisfy the equation 



S'(x) = {\R + B) S(x) + 4 W(*) " B(x) P k (x)} E(x), 



A 



i.e. an equation of the form 



(32) S'(x) = | \R(x) + B(x) + 1 1(;r, X) j S(z), 



where Z(x, X) is a matrix each element of which is rational in X, 

 with coefficients continuous in x, given by power series in ( - ). 



If on the other hand k = °° , as many terms of the infinite series 



~fl> J?) 



