76 BIRKHOFF AND LANGER. 



Since the other quantities under the sign of summation in this formula 

 have all been determined, pjj may be found by means of a quadra- 

 ture. When this has been done the matrix Pi has been completely 

 determined. 



It should be observed that we have thus far required only that the 

 matrices R(x) and B(x) be continuous. To determine the elements of 

 Po however, we have first, for i =(= j, the equation 



(2) — Pij + L "ik pkj 



Pa - k=i y 



7™ Ti 



and since this formula implies the existence of pij, whereas it is seen 

 from (29) that p® has a derivative only if this is true of the matrices 

 R(x) and B(x), we cannot proceed to the determination of P> if R(x) 

 and B(x) are merely continuous. 



Let us suppose then that R(x) and B(x) both possess continuous 

 derivatives up to and including those of order h ^ 1, but that perhaps 

 one of these matrices possesses no such derivative of order (k + 1). 

 If, in particular R(x) and B(x) possess infinitely many derivatives we 

 may take h = oo . The derivatives of Pi up to and including that of 

 order k are now seen to exist from formulas (29) and (30). 



Equating formally the coefficients of -— [ in the identity (27) we 



A 



have 



i.e. 



1\R + iVi = iU\ + BPp-i, 



n 



Pij 17" 7»J — —pij t- OikPkj > 



A-l 



n 



Pij + - °ik Pkj 



whence p^ = — for i =f= j, 



7/ - 7; 



1 

 and again, equating the coefficients of— we have 



A 



P M+ i R + P' M = PP M+ i + BI\ 



