190 PROCEEDINGS OF THE AMERICAN ACADEMY. 



is then 2 m. Similarly, if p is any positive integer, and if tte 

 nullity of 



(^ _ ^3T ^p = Q (^- 1 _ ^zn: ^)P Q-i 



is m, the nullity of 



is also m ; and therefore the nullity of {ip + V — 1 S)^ is ?«. But then 

 the nullity of 



is 2 m. 



We have therefore the following theorem by which we may ascer- 

 tain the existence of substitutions of the second kind. If ^ is a linear 

 substitution of the first kind which satisfies the equation 



</) f2 (^ = i2, 



then if — 1 is a root of the characteristic equation of <f), the nullity of 

 any positive integer power of ^ + 8 is even. 



That substitutions of the second kind actually exist may be shown 

 by considering the form 



« (^i Vx — ^'2 y-i) + ^ ^2 y\^ 



which is transformed automorphically if we impose upon the a:'s the 

 substitution whose matrix is 



-1 1 



-1 



and upon the ^s the transverse substitution ; that is, if we put 



3"l = — Cl "T ^25 X2 ^ t2» 



and 



3/1 = — y]\-, 1)2 = rix — V2- 



This linear substitution does not satisfy the preceding conditions, 

 and is therefore of the second kind. It follows that there are one or 

 more bilinear forms for any number of variables which are trans' 

 formed automorphically in the manner we have considered in this 

 section by linear substitution of the second kind. 



By definition, no solution of the second kind of the equation 



