Determinant Notation. 30 ( J 



to the /x rows 9 and the v columns <f> of the array 



(<*«) (p= l , .••»<; 7=1, ...») 



may be denoted by (a e<p ). In particular the array formed by 

 all the elements in the /j, rows 6 may be denoted by (ct eq ). 



With this notation Sylvester's second theorem may be 

 stated as follows : — 



If 0, </> be i/-ads from 1, . . . n and 



then 



Eflr* = 



Efl,-n I = 



(Clpq) {b p( p) 



(c eq ) (dee) 



(p, q=l,...m), 



J e$ 



(<*pq) ( f 'ps) 



{Crq) [d n ) 



I A' 



(r, S=l, 



rhere 



X = 



x' = 



(w-l)(n-2) . . . (n-v + 1) 

 1.2 ... v 



(n-l)(n-2) . . . (n-v) 



1:2 



(v-1) 



Proofs of this theorem have been given by Reiss*, Picquetf, 

 Scott $, and Van Velzer §. Scott deduces the second theorem 

 from the first in the following simple way : — 



By the first theorem we have 



where A = 



Hence the determinant 



which is of order X + X', is the v-th compound of | c rs | and 

 is therefore, by Franke's theorem, equal to | e rs \ K '. But by 

 the first theorem we have j e rs | =A re_1 A, and hence we 

 readily find | E^ | = AW. 



Van Velzer remarks that this proof " seems to leave nothing 

 to be desired, either in simplicity or rigour." Sylvester's 

 second theorem is, however, merely the extensional of Franke's 

 theorem already quoted. It is certainly remarkable that a 

 very general theorem enunciated without proof by Sylvester 



* Beitrage zuv Th. der Bet. p. 34 (1867). 

 t .Journal de VEcole Pol. Cah. xlv. p. 216. 

 X Proc. Loncl. Math. Soc. xiv. p. 93. 

 § Am. Journ. Math. vi. p. 168. 



