TABER. — TRANSFORMATION OP A REAL BILINEAR FORM. 313 



The characteristic equation of a transformation of V of the first kind 

 may have negative roots. But in such case the numbers belonging to 

 each of the negative roots of this equation are all even ; and therefore, 

 for each negative root of this equation, the elementary divisors (elementar 

 Theiler) of the characteristic function corresponding to such root occur 

 iu pairs with equal exponent.* 



No transformation of V with negative determinant is of the first kind. 

 Let n z=. 2, and 



|o,il 



The form Jp is transformed automorphically if 



(xi', xJ) = T{xt,, X2) = {—I, 1^X1 x^, 



I 0,-11 



(y/, y,<) = f-' {y, , y.^ = (- 1, O^yi , y,). 



1-1,-11 



"We have I T \ ■= + 1, but 7' is a transformation of the second kind. 

 Whence it follows, for any value of n, that there are forms J such that T' 

 contains transformations of the second kind with positive determinant.! 



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

 of a transformation of T' ; but every transformation ofV'of the second 

 kind is the (2 m + Vf^ power, for any odd exponent 2 m + \, of a trans- 

 formation of T' . Thus, let T be any transformation of V, and, as before, 

 let Fand IF be real polynomials in 2' satisfying the equation 



As shown above, for any real scalar ^, e^^ is a transformation of T' ; and, 

 therefore, so also is e^^~^ = Te~^. Consequently 



* For the roots of the characteristic equation of T are the squares of the roots 

 of the characteristic equation of T; and, if T is real, each imaginary root of tiie 

 characteristic equation of T is paired witli its conjugate imaginary. Compare 

 These Proceedings, 31, 189. 



For the relation between the numbers belonging to the roots of the character- 

 istic equation of a linear transformation T and the exponents of the elementary 

 divisors of the characteristic function of T, see Bull. Am. Math. Soc, 2d series, 

 3, 156. 



t The condition, given on p. 308, sufficient that r' shall contain transformations 

 of the second kind, is readily obtained. 



