582 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



Let cj),, denote the matrix of the bilinear form —2^2^(2 (t-j Cj v ^) y^ z,, 

 namely, 



/— 2a^-(?^.ii, — 2ajCj.2i, . . . — 2aj-c^-,i 



^ Oj Cj 12, — 2 (Ij Cj 22» • • • 2 tlj Cj J. 2 



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

 denote the series I + 4> „ + h 4>a' + • • •> which is convergent for any 

 matrix (/>„, Then it will be found that 



■^115 -^12? 

 -^21) -^22) 



. . AJ =— I = /+ *</>„ + 



\ 



^a 



Let now A„ denote the determinant of 



/-7 



^a 



This determinant 



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



2 Clj Cjii — 2 k-n- V — 1, 2 Ctj Cyi2 , • • • 



2 aj Cji2 , 2 cij Cj„o — 2 A" TT V — 1 • 



0, 



where k is some integer not zero. The values of the parameters a for 

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

 The critical values of the parameters a 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 tt ^y/ — i. 



If A„ =}; 0, we may take the b's arbitrarily, and then, from equations 

 (12), derive expressions for Sa-i^Sa^, . • • 8 «,., as linear functions of 

 ^1, b.^, . . . b,.. Thus, if A„ 4= 0, we have 



x'i = 2k hiv Xi (i =1,2,... n), 

 1 



A ±. B denotes the matrix of the linear substitution 



x'i = 2" {aiv ± bh) Xi (i = 1,2, . . . n), 

 1 



and A B the matrix of the linear substitution 



tt n 



x'i = 2*^ 2>/ «)> b^v Xv {i = 1, 2, . . . n). 

 1 1 



We shall then have A {B C) - {A B) C, A {B ± C) -A B + A C, etc., but in 

 general AB :^ B A. 



