820 PROCEEDINGS OF THE AMERICAN ACADEMY. 



JF = (0, 1 §x u zsfefx, 3/ 2 ) = (*i y 2 + * 2 y x ) + bx 2 y 2 . 



The form jf is transformed automorphically if 



(xj, x.>) = (7*8*!, * 8 ) = (-1, %x, * 2 ) 



|o,-i| 

 I 0, — ij 



We have 17*1 = +1, but T is a transformation of the second kind. 

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

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

 determinant.* The form 



J = (—a 2 , <%!, x 2 fy/ u y 2 ) = — a 2 x 1 y 1 + x 2 y z 



I 0, 1| 



is transformed automorphically if 



(*/, x 2 ') = (T$x u x 2 ) — (—1 + a A, A 8*!, a: 2 ), 



—a 2 A, —1 -a\\ 



<J/i, y 2 ') = ( Z*-%i, y 2 ) = (-1 - a A, -A fo, y 3 ) ; 



a' 2 A, — 1 + a Af 



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



* °" ' 



Every transformation V" of the second kind is the (2m + \) ttl power 



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

 be any transformation of T'". If Fand IF are real polynomials in T 

 satisfying the equation T = e F+irV_1 , we may then show, precisely as 



— — F+wV— 1 

 for the similar theorem in the case of the family T" that e 2m+1 is 



a transformation of Tf'\ and that 



T= e r+wV ^ = ( e ^i F+>rV ^ i ) 2m+1 .t 



Clark University, Worcester, Massachusetts. 



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

 kind see p. 308. 



t See p. 313. The family of transformations V" also constitute a group. 



