582 



■ EEDINGS mk THE AMERICAN ACADEMY. 



Let <f> a denote the matrix of the bilinear form — - M -, (5 " c , M ) v M :>■ 



i i i 



namely, 



/ V 



2a,e, u , — S%e in , . 



" — " i c i LSI — ", '', IS) • 



— - «j Cjr 1 



— "( '"; Irt — "., '') ^ r« 



^ il r 



I J i r ! 



Let /denote the matrix unity (the identical transformation), and let e*" 

 denote the series / + <£ „ + A <f> a * + . . ., which is convergent for any 

 matrix <£„. Then it will be found that 



fA u , A V1 . . . . AA e * a _ j 



.l.j!, A.,.,, . . . AnJ 



</>" 



= /+i^.+ . 



Let now A a denote the determinant of 



4> a 



vanishes it' and only if the a's^are so chosen that 



2 <ij Cm — 2 kir V — 1, 2 a, Cjn 



This determinant 



= 0, 



za t c 



j l >12 



, 2 a j Cj.,., — '2kTr\/— 1 



where k is some integer not zero. The values of the parameters a f<>r 

 which A„ vanishes may be termed critical values of the parameters. 

 The critical values of the parameters u are, therefore, those values of 

 the a's for which one or more of the roots of the characteristic equation 

 of the matrix <£„ is an even multiple, not zero, of it y/~^\. 



If A„ 4= 0, we may take the i's arbitrarily, and then, from equations 

 (1 2). derive expressions for 8a u 8tr„, . . . 8«,., as linear functions of 

 b 1} b,, . . . b r . Thus, if A„ 4= ^- we have 



2, b v .r, (» = 1, 2, . . . n), 



l 



A ± B denotes the matrix of the linear substitution 



x'i= S" (a, v ±l,v)n (»= 1,2, 



i 



and A B the matrix of the linear substitution 



»), 



S, = 2^ 2v (Zip 6^ *» (' = li 2, . . . n). 



l l 



V7e shall thm have .1 [B C) = (.1 B) C, A (B ± C) = A B + A C, etc., but in 

 general .1 B '.. BA. 



