182 PROCEEDINGS OF THE AMERICAN ACADEMY. 



contragrediant variables can be generated by the repetition of an infin- 

 itesimal substitution of the group. For, if the two sets of variables 

 of the bilinear form 



are contragrediant, and if 



(a.-i, To, . . . X,) = (<^ 55 tl' ^2, • • • Q, 



(rji, rjo, . . . 7j„) = {f^yui/.„'-- y,), 

 in which ^ denotes the transverse or conjugate to ^, we have 



(.^-1 O (^ ^ t^l, t^2, . . . 4 ^ 7?!, 770, . . . 77,.) 



= (fi ^ ari, 0-2, . . . ar„ i;^ yi , 2/2 , . . . y„) ; 



and the necessary and sufficient condition that the transformation shall 

 be automorphic is 



or 



fi(/) = </)f2.* 



Let now 8 denote the identical substitution. Since it is assumed 

 that a reciprocal of </> exists, that is, | | =^ 0, a polynomial ;^ = y (^) 

 in ^ can be found such that 



<^ = e^ 



* In this paper I employ the notation of Cayley's " Memoir on the Linear 

 Automorpiiic Transformation of a Bipartite Quadric Function," Philosophical 

 Transactions, 1858, with these exceptions, namely, the identical substitution will 

 be denoted by S, and the linear substitution or matrix transverse or conjugate 

 to the linear substitution or matrix ^ will be denoted by ^. Cayley denotes the 

 bilinear form 2,- 2s ar,Xsjjr {>', s = 1, 2, . . . n), as above, by 



(n (J xi, xo, . . . x„^ yi, y2> ■ • ■ yn), 

 the symbol XI denoting the matrix 



a.2i a.22 . . . 



that is, the square array of coefficients of the form. 



The determinant of the linear substitution (p will be denoted by | <j) | 



