320 PROCEEDINGS OF THE AMERICAN ACADEMY. 



J = (0, l()xi, Xo^^i, 3/2) = (^1 2/2 + a^2 2/1) + i a;2 ^2. 

 11, ^1 



The form j]F is transformed automorphically if 



(x/, a:/) = (T^xi.x,) = (-1, 1^X1, x^) 



|o,-i| 



(2/1', ^2') = {T-'lVvV^) = (-1. -l^yi'2'2). 



I 0,-l| 



"We have jJ'! = +1, but J' is a transformation of the second kind. 

 Whence it follows, for any value of n, that there are bilinear forms JF 

 such that r'" contains transformations of the second kind with positive 

 determinant.* The form 



iF = (— aS O^xi, X2^yi, 1/2) = — a^ ^i Vx + a^2 Vz 



I 0, 1| 



is transformed automorphically if 



(x/, xJ) = (T^x^, X2) = (-1 + aX, X l^x^, X.2), 



1 — a^A, — l-aA| 



(j/i, W) = ( ^"'^^1' 2/2) = (-!-« X, -X p,, 2/2) ; 



I a^X, —l + aX\ 



and T, of determinant +1, is a transformation of the second kind if \ 



to- 



Ecery transformation V" of the second kind is the (2 m + 1)''' power 

 of a transformation of V'" for any odd exponent 2m + 1. Thus let T 

 be any transformation of T"\ If Fand FFare real polynomials in T 

 satisfying the equation T = e^+'^*^\ we may then show, precisely as 



for the similar theorem in the case of the family T' that e^"'+^ is 



a transformation of T"\ and that 



Clark University, Worcester, Massachusetts. 



* For a sufficient condition that r'" shall contain a transformation of the second 

 kind see p. 308. 



t See p. 313. The family of transformations r'" also constitute a group. 



