BOUNDARY PROBLEMS AND DEVELOPMENTS. 81 



apparent, therefore, that the solution associated with (37) is of the 

 form 



(38) T(x) = («rxri<»>-itf(«) [5..]). 



Section VI. 

 The relation of the formal solutions to the actual solutions. 



It was observed in the preceding section that the formal matrix 

 S(x) either satisfies equation (26) only in an approximate sense (i.e. 

 satisfies (32)) or, if k = °° satisfies it only formally, since the elements 

 of S(x) are in that case infinite series which are not in general con- 

 vergent. S(x) is, therefore, not a matrix solution of equation (26), and 

 its significance requires further investigation. 



Consider the actual matrix which is derived from S(x) by retaining 

 in the latter only the first (m + 1) terms of its elements, where m is any 

 positive integer not exceeding k. This matrix S(.r) may be written, 

 in accordance with formula (37), 



S( x ) = (,*r ; (*) + B ; (*) [g..]j 

 = P(x, X) E(x), 

 where P(x,\) = (e B W [8 i3 ] m ), 



and is seen to be analytic in X. Since | f s > (x) 5{ ; - 1 =(= 0, it follows that 

 I iS I =4= for I X I > N. If the formal solution S(x) is written as 

 P(x, X) E(x), it is apparent that we have the formulas 



Six) = \l - ^ [Q] P" 1 J S(x), 

 S(x)= J 7+ ~{Q}P- l ]s(x), 



from which we obtain upon differentiating and substituting from the 

 equation (32), the relation 



