TARER. — TRANSFORMATION OF A REAL BILINEAR FORM. 317 



Every transformation of T" with negative determinant is of the second 

 kind. Let n = 2, and let 



J = (1 , Ofix lt x$y x , y 2 ) = x t y t — x 2 y 2 . 



i.o, — i r 



The form j]F is transformed automorphically if 



(*i'» a?a') = T(x x , ar 2 ) = (— 1, 1^, z 2 ) 



I 0,-11 



W, fj) = f(y u y,) = (- 1, <% lf yO- 



1 1,-H 



We have I T 7 ! = + 1, but T 7 is a transformation of the second kind. 

 Whence, for any value of n, it follows that there are forms JJ such that 

 T" contains transformations of the second kind with determinant +1.* 

 Again, the form 



J E E ( 0, 0, 



o, o, 



a. 



b, d garj , x , x 3i x^y 1} y 2 , y 3 , y 4 ) 



0, — b 



0, 



0, =a (x l y z — x 2 y 4 ) + b (x z y x — x^ 2 ) 



+ cx 2 y 3 + dx. v y x 



is transformed automorphically by the transformation 

 T. 



0, -A 2 , 

 0, 0, 

 0, 0, 



0, ) 

 0, 

 A- 2 , A" 4 

 0, -A" 2 



of r" ; and 7* is a transformation of the second kind if A. =|r ± 1. 



By definition no transformation of the second kind is an even power 

 of any transformation of T" ; but every transformation of the second kind 

 is the (2 m + \ )' h poioer, for any odd exponent 2 m +1, of a transforma- 

 tion of T". Thus let T be any transformation of T" ) and, as before, 

 let f(T) be a polynomial satisfying the equation T= e^ 1 , and such that 

 f(T) —f(T~ v ) is real. Then, as shown above, if 



2 Uo =f(T) +/(7 7 - 1 ), 2 L\ =f(T) -f(T-% 



* The condition, given in note on p. 308, as sufficient that T" shall contain 

 transformations, is readily proved. 



