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 



JJ = (^ (^ xi, a;2, . . . ar„ ^ y^, y^, . . . y„) 



denote a bilinear form 2" 2" a„ x^ y^ of non-zero determinant. Let 



11 

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



transformations T'and 7\, so that 



(Xi ) Xn , . . . X,^ ) = {^I i^ Xi, ^2, . . . Xj^) , 



(yi, y-i, • . • yJ) = (^i ^ ^i^ x^, . . . x,) . 



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

 formed automorphically, so that 



\~ Q "^1 5 '''2 3 • • • x,^)^=- {A. Q a?i, X2j . . . x^) , 



for which the necessary and sufficient condition is 



T^A T=A* 



I shall denote the family of transformations T of the x^s by 



1st, r', when the x^s and y^s are contragredient, in which case we have 



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



(J 



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



3d, by r'", when the product of the respective transformations of the 

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

 The conditions necessary and sufficient that T shall be a transformation 

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



* Cayley : Pliil. Trans., 1858. Tliroughout this paper f will denote the trans- 

 verse or conjugate of the linear transformation T. 



