ON THE REAL AUTOMORPHIC LINEAR TRANSFOR- 

 MATION OF A REAL BILINEAR FORM. 



By Henry Taber. 



Presented October 14, 1903. Received October 22, 1903. 

 § 1. 



Let 



J = (A Q X U * 2 , . . . X n 5 y x , y a , . . . y B ) 



denote a bilinear form 2" 2" a rs x r y s of non-zero determinant. Let 



11 

 the x's and y's be transformed respectively by the linear homogeneous 



transformations T'and T x , so that 



(Xi , Xn , . . . x n ) = ( 7 Q a: 1? a: 2 , . . . a? n ) , 

 (fr'tSta't • • • y»0 = C^i^u »«» • • ■ K ») • 



It will be assumed as the result of these substitutions that $ is trans- 

 formed automorphically, so that 



{A Q x x , x 2 , . . . x n ) = {A Q x u x 2 , . . . x n ) , 



for which the necessary and sufficient condition is 



T X A T=A* 



I shall denote the family of transformations T of the x's by 



1st, T', when the x's and y's are contragredient, in which case we have 



25 = 1M ; 



2d, by r", when the respective transformations of the x J s and of the 



KJ 



y's are conjugate, so that T\ = T; 



3d, by T'", when the product of the respective transformations of the 

 x's and of the y's is the identical transformation, so that T x — T~ x . 

 The conditions necessary and sufficient that T shall be a transformation 

 of r', r", or r"', respectively, are then 



* Cayley: Phil. Trans., 1858. Throughout this paper T will denote the trans- 

 verse or conjugate of the linear transformation T. 



