184 PROCEEDINGS OF THE AMERICAN ACADEMY. 



whence it follows that any linear substitution of the group of linear 

 substitutions which transform automorphically a bilinear form with 

 contragrediant variables can be generated by the repetition of an 

 infinitesimal substitution of the group. 



§2. 



If the two sets of variables of the bilinear form 



are transformed by linear substitutions transverse or conjugate to each 

 other, so that 



(a:i, x^, . . . a:,,) = (<^ ^ ^i, ^2, • • • L), 



(j/uy^, ' ■ ■ y„) = (i^Vi^Vi^ • ' ■ Vn), 



where </> denotes the linear substitution transverse or conjugate to <^, 

 we have 



(^n<t>^$i, |o,.. . ^„^ 771,770, . . .r]„) 



= (n^Xi,x., . . .x„'^yi,yr,, . . . y„). 



The necessary and suflFicient condition that this transformation shall 

 be automorphic is that <f> shall satisfy the equation 



The class of linear substitutions that satisfy this equation, that is, 

 the linear substitutions which transform the bilinear form in the man- 

 ner described, do not form a group ; but they can be separated into 

 substitutions of the first or second kind according as they are or are 

 not the second power of a substitution of this class.* And any sub- 

 stitution of the first kind can then be generated by the repetition of 

 an infinitesimal substitution of this class, whereas no substitution of 

 the second kind can be generated thus. 



* If <pn.(p = a, then <p- a (p"^ = <p {<p a (p) (p = <p a <p = n. 



