662 PROCEEDINGS OF THE AMERICAN ACADEMY. 



m 



In this case, there is some number A = YL ('i^i 7^ of the system 



m 



(ei, 62, ... Cfn) such that A X = for any number A' = J^ .r,-?t of 



1=1 



this system; since we should then have 



TO TO TO 



AX = L L L Oi^jJijkek 

 t=l y=i k=l 



mm TO 



= Z Z (Z (Hyijk)xjek= 0. 



TO 



Conversely, if ^i = Z cuei^O and /I A^ = for every number A' of 



»=i 



(ei, 62, ... e,„), equations (89) are satisfied for at least one system of 

 values Ui, 02, . . . am not all zero, and we cannot assign to the d's, nor 

 to the rj's, the values given by equations (88). In this case, we have 



SiAei = = S'lAci {i = 1, 2, . . . m); 



and, therefore, 



Ai = Ao = 0. 



It is to be noted that equations (89) are the conditions, necessary 

 and sufficient, that the reciprocal system shall contain a number 



TO TO 



A' = J^ iiie'i 7^ such that A''. 4' = for any numl)er A'' = Z ^i^'i 



i=i " t=i 



of this system. 



Further, we may take n = m and put 



(90) duv^^ = Turn Vui'^ =Twu {i, u, V = 1,2, ... m), 

 unless for 61, bo, ... b^ not all zero, we have 



TO TO 



(91) Z ^iTui. = Z ^if^u.^'^ = 



(u, V = 1, 2, ... m), 

 in which case Ei, E2, ... E^ are not linearly independent, nor are 



