Mr Bromwich, The Classification of Conies and Quadrics. 359 



1. Preliminary consideration of bilinear forms. 

 Let A = %a rs a r y s , B — %b rs x r y s be two bilinear forms each of 



r, s r,s 



which is symmetrical, so that a S) . = a rs , b sr = &,,,. Then let 

 C = Xc rs x r y s = \A — B, 



r,s 



so that C is also a symmetric bilinear form, and then consider the 

 form 



where 



D = K 



A,= 



(r,s=l, 2, ...,n), 



Here the p's are supposed to be constants and the deter- 

 minants are abbreviated symbols ' for determinants of (n + 2) 

 or (n -f- 1) rows and columns. 



Take the symbolical product CD in the sense given by 



Frobenius (Crelle, 84 (1878), p. I) 2 ; this is obtained by writing 



Xctr&t instead of x r in D. Thus 

 t 



CD = 



1 



Crs , Pr, IJi 



Ps , o, 



%c ts oct> , 



1 



a; 



Crs, Pr, l/r 



Ps, 0, 



, -tprOCr, -E 



where E = %x,.y r is the unit-form {Einheitsform). 



It will save a certain amount of repetition if we write 

 p z = Sp, r x r , p y = 2p r y r . 

 With these symbols we now have 



CD = -E + 



Px 



Crs, Vr 

 Ps, 



1 This suggestion has been made both by Frobenius and by Nanson. 



2 This method of multiplication is really Cayley's method of combining 

 matrices. 



2G-2 



