TAKER. — TRANSFORMATION OP A REAL BILINEAR FORM. 317 



Every transformatiou of r" with negative determinant is of the second 

 kind. Let n = 2, and let 



Jf=(h 



^2^yi} Vi) = a^i^i — ^^Vi- 



I, O^xi, 



10, -ir 



The form J is transformed automorphically if 



(a;/, x^) = T(xi, x^) = (— 1, l^x^, x^) 



I 0,-11 



Q/i', y2') = T(y„ y,) = (- 1, 0(5yi, y,). 



1 1,-11 



We have | 7^1 = + 1, but J' is a transformation of the second kind. 

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

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

 Again, the form 



dl^xi, X , xs, 3c.i^yi,y2,yz.y^ 

 -b 

 



= a (a^iya — Xo y^) + b (x^y^ — x^y.^ 



+ cx^y^ + dx^yx 



is transformed automorphically by the transformation 



T^{-\\ 1, 0, ) 



0, —X\ 0, 

 0, 0, -A-2, A-* 

 0, 0, 0, -A-2 



of r" ; and T'is a transformation of the second kind if A rj: ± 1. 



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

 of any transformation of F" ; but every transformation of the second hind 

 is the {2 m 4- I)"' poicer, 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 2'= e^^\ and such that 

 f{T) —f{T-^) is real. Then, as shown above, if 



2 U, =f{T) +f(T-% 2 U, =f{T) -f(T-% 



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

 transformations, is readily proved. 



