BOUNDARY PROBLEMS AND DEVELOPMENTS. 



73 



(23) 



2 {o ik — 8ik7j a } <Pkj a = 0, 



obtained by giving to j successively the values 1, 2, . . n. This 

 means that each of the determinants | a ik — 8i & 7/ |, jo = 1,2,. . .n 

 must vanish, and inasmuch as the functions 7/(.x) were chosen as the 

 roots of equation (22), which is precisely the condition that the de- 

 terminants in question do vanish, the existence of a matrix <p(.r) of the 

 type desired is established. 



Condition (ii) requires further, however, that this matrix $ be one 

 for which | 4>(a:) | 4 1 0. 



Suppose | <£(.r) | = 0. Then the system of linear equations 



2 (fik Vk 



= admits of solution by means of a set of quantities V\, 



v 2 ,... v n , at least one of which differs from zero, i.e. there exists a vector 

 V • ^ 0, such that 3>F- = 0. But then A$V- = 0, and in view of the 

 relation (i) it follows further that $i?T'- =0. By continued repeti- 

 tion of this reasoning it can be shown likewise that &R V • = for all 

 values of k. Consequently 3>{c I + c i R+c 2 R 2 + . . . -f c n _i iT" 1 } V-=0. 

 But on the hypothesis that 7, 4= li when j ^z i it is well known that 

 the determinant 



1 2 



1 7i 7i 



1 72 72 2 



■7i 



■72 



re-1 



ra-1 



■7n 



n-1 



+ 0. 



1 7n 7n 2 



Hence the system of equations 



Co+('i7i + (VYi 2 + • ■ ■ +C«-l 7i n_1 = &1 



f o+ Ci7„ -f- c 2 7„ 2 + • • • + C„_i 7i n = k n , 



admits of a solution in the c's not all zero for any choice of the quanti- 

 ties ki not all zero. Some choice of the set c , C\. . .c n _\, can therefore 

 always be made to satisfy the relation 



{c J + ci R + . . . + c n _! R n ~ 1 } = (Sa ki). 



8 Cf. Bocher, loc. cit., pp. 47. 



