Dr Muir on ProUem of Sylvester’s in Elimination. 303 
the dialytic method cannot he applied to the six equations (a) and 
(y). In fact, these sets must, for dialytic purposes, be looked on as 
identical. 
5. But now taking (/?), and eliminating xy, yz^ zx by means of 
(ttj), (a 2 ), (ttg), we obtain 
A'C(A'C'-BBV + B'C(B'C' - AA')y2 + C'(AA'2- 2A'B'C' + BB'2)22 = 
Similarly 
A'(BB'2-2A'B'C' + CC'V + B'A(B'A' - CC'y + C'A(C'A' - BB')s2 
and 
A'B(A'B' - CC')x2 + B'(CC'2-2A'B'C' + AA'%2 + C'B(C'B' - AA>2 
From these we have 
C(A'C' - BB') C(B'C' - AA') AA'2 - 2 A'B'C' + BB'^ 
BB'2 - 2A'B'C' + CC'2 A(B'A' - CD) A(0'A' - BB') 
B(A'B' - CD) CC'2 - 2 A'B'C' + AA'2 b(C'B' - A A') 
or, multiplying the columns by A, B, C, and dividing the rows by 
C, A, B respectively, 
A(A'D - BB') B(B'C' - AA') AA'2 _ 2A'B'C' + BB'2 
BB'2 - 2A'B'D + CC'2 B(B'A' - CD) C(DA' - BB') 
A(A'B' - CC') CC'2 - 2A'B'C' + AA'2 c(c'B' - AA') 
ithe determinant in which is equal to 
C'2-AB 
AA' - B'C' 
BB' - C'A' 
B' 
A' 
BB'-C'A' 
CC'-A'B' 
A'2 - BC 
X 
C' 
A' 
AA' - B'C' 
B'2 - AC 
CC'-A'B' 
C' 
B' 
that is. 
A2 
X ( - 2A'B'C') . 
6. Next, taking {(B-^ and (yg), and eliminating we obtain a 
set of equations involving only yz, zx, xy, viz., 
them by changing the letters in accordance with the indications of the cycles 
O' 
—a mode of derivation that naturally applies to the equations of every other 
set likewise. 
