1 



X 



m* 



X 3 



a* 



a 



b 



c 



d 



c 



a 1 



y 



c' 



cV 



c' 



a" 



i" 



c" 



d" 



e" 



a"' 



h w 



c'" 



d"> 



e'" 



456 Prof. Sylvester on the Theorems 



and 



x 4 = 



Or, according to a suggestion of Mr. Cayley, putting the question 

 into a more general and simple form, we may inquire into the 

 resultant of any two complete determinants, functions of x of the 

 ?zth degree, which belong to the rectangular matrix 



1, X, X 2 ... x n ~ l 3 



#1,1; a l,2) #1,3 • • • Q\,nj 



#2, \3 #2, 2) #2, 3 • • • 0-2, nj 



a n-l,D &7i-l,2) a n—l,3»'- a n-l,m 

 #», 1) #rc, 2) #», 3 • • • #n, n> 



as, for instance, the resultant of the two determinants which may 

 be obtained by suppressing successively the last and last but one 

 line in the matrix above written : and by aid of the most ele- 

 mentary principles of the calculus of determinants the instructed 

 reader will find no difficulty in proving that this resultant will 

 resolve itself into two distinct parts — one a power of the determi- 

 nant obtained by suppressing the uppermost (or x) line in the 

 above matrix, the other the Resultant of the matrix obtained by 

 suppressing simultaneously the two lowermost lines*. This last 

 suppression leaves a rectangular matrix which, written in a ho- 

 mogeneous form, becomes 



y n -\ y n ~ 2 .x, y n ~ 3 .x 2 ...x n ~ l , 



#1,1; #l,2j #1,3 • • • a \,n> 



#2,1 j #2,2j #2,3 • • • <h,W 



a n-2,l> a n—2,2> #n-2,3 • • • a n-2,n) 



consisting of n columns and (ft — 1) linesf. 



* For, on making the last-named determinant referred to in the text zero, 

 it may easily be shown, by aid of a familiar theorem in compound determi- 

 nants, that the two determinants whose resultant is under investigation have 

 all the coefficients of the one in the same ratio to each other as the corre- 

 sponding coefficients of the other. 



\ The reader may notice that the real interest of the subject under con- 

 sideration commences with the independent inquiry into the form of the 

 Resultant of the above matrix — the original question, as to the quasi-cano- 

 nizant, being important only as leading up to the appearance of this Resultant. 



