Ai = 



EXPANSION PROBLEMS. 397 



wi^"!!] 1 f W2*^''e'"^(''^"") [1] \ 



+ a,p^-^''wi''c-''"'"^[l] J \ +a«p''-^>W2''e-''"'"^[l] J ■"■ 



/ w/j'ePT(M'i'~«'i) [1] 



the exponent k — /.> in the last row of this determinant may be posi- 

 tive, negative, or zero. 



It is at this point that the import of the remarks made at the close 

 of the last paragraph but one becomes apparent. As p becomes infi- 

 nite in the region Tq, the first element in the last row of Ai approaches 

 Uiiiformly the limit Wi^^, and the elements from the third to the last 

 uniformly approach zero. The second of the two terms which make 

 up the second element of the row is also uniformly infinitesimal. 



Let us expand the determinant according to the elements of the last 

 row, and examine the cofactor of the first element. If we suppose 

 each bracket symbol replaced for the moment by unity, the cofactor 

 is a determinant whose value is surely different from zero, not because 

 its rows consist of like powers of a set of distinct quantities (it is not 

 assumed that all the exponents kg are sxccessive integers), but because, 

 after division of the s-th row by ^^2*^ s = 1, 2, . . ., i/ — 1, an operation 

 which of course does not affect the vanishing of the determinant, each 

 column may be regarded as consisting of like powers of the v — 1 

 distinct quantities iVi^'"'^, iV'f^'^, . . . , loi'^v-i^ the exponents of these 

 powers in the several columns being the integers from to v — 2 

 inclusive, in some order; the numbers Wz/ W2,. ■ .w^/ w^, are even 

 powers of w^. If the brackets are restored, the cofactor is a function 

 which uniformly approaches a limit different from zero when p becomes 

 infinite. Similar reasoning, with a like conclusion, applies to the 

 cofactor of the second element. Representing the limiting values of 

 the cof actors of the first and second elements by 5i and ^2 respectively, 

 we see that A] has the form 



(19) Ai = [5i wi^^] + [52 w-^^] e "'^^ ""'=-"'1) ; 



the terms in the expansion which involve the last v — 2 elements 

 and the second part of the second element have been merged here in 

 the first bracket symbol. 



If A2 is the expression which results when the bracket expressions 

 in (19) are replaced by their limiting values, the equation A2 = can 



