BOUNDARY PROBLEMS AND DEVELOPMENTS. 75 



(27) |po+^Pi+j~P2+... j \re+ |p;+^p;+ ... j E = 



jxp + pj ^Po+ 1 -p 1+ ..l E . 



Equating the coefficients of X we obtain the relation 



P R = PP 0) 



(°) (0) 



whence it is seen that pfj = 0, when j 4= i. Equating the coefficients 

 of X we obtain in similar manner the equation 



P r R + Po' = PPi+ PPo, 

 i.e. 



(28) pS ) 7i + l$ ) ' = 7<i»8 ) + 2 &«rf?, 



whence it is seen, upon setting i = j that 



n 



„,(0)' _ y J. (0) __ j (0) 



Pjj - ^ b ik Pkj - Ojj Pjj , 



fc-1 

 x 

 J bjjdx 



namely, that p)j — /,- e a 



Having obtained in this manner all the elements of the matrix P 

 we find further from the equation (28) for i =£ j that 



(29) pg> = ^ • 



7;— li 



Moreover, equating the coefficients of - we see that 



A 



P 2 R + P[ = RP*+ BP U 

 i.e. 



it=i 

 whence it follows upon setting i = j that 



(i)' _ v ;. AD 



(30) p}r = ?b 3kP % 



fc-1 



