302 Proceedings of Poyal Society of Edinhurgh. 
BC 
. 
AA' 
-BB' 
-CC' 
. 
CA . 
- AA' 
BB' 
-CC' 
. AB 
-AA' 
-BB' 
CC' 
A' 
A 
-C' 
-B' 
B' 
-C' 
B 
-A' 
. 
C' 
-B' 
-A' 
C 
where, of course, the determinant is no simpler than Sylvester’s. 
That it leads to the same result is easily made clear by first multi- 
plying the 4th, 5 th, and 6th rows by BC, CA, AB respectively, 
and then dividing the first three columns by BC, CA, AB respec- 
tively. We thus obtain 
1 
ABCx 
A' 
thence 
ABCx 
1 . . 
. 1 . 
. . 1 
A' 
-B' 
-C' 
. -A' 
B' 
-C' 
1 -A' 
-B' 
C' 
BC 
-CC' 
-BB' 
. -CC' 
CA 
-AA' 
C' -BB' 
-AA' 
AB 
A' 
-B' 
-C' 
-A' 
B' 
-C' 
-A' 
-B' 
C' 
BC - A'2 
A'B' -CC' 
C'A' - BB' 
A'B' -CC' 
CA - B'2 
B'C' - AA' 
C'A" -BB' 
B'C' -AA' 
AB -C'2 
and finally 
ABC 
A C' B' 2 
C' B A' 
B' A' C 
4. It is of importance to notice that since any one of the equations 
(y) may also be got from the equations of the original set (a) by 
mere addition or subtraction of multiples, e.g.^ 
7i = i( - + Aa^ - Btt3),* 
* Of course there are similar operational equations for the obtaining of 
and 73 , because any one of the original equations is derivable from another of 
