418 
Proceedings of Royal Society of Edinburgh. [sess. 
Ay 2 - 20! xy + Bx 2 = 0 (1), 
Bz 2 - 2A!yz + Cy 2 - 0 (2), 
Cx 2 - 2B 'zx + A z 2 = 0 (3). 
Multiply the equations in order by - z 2 , x 2 , y 2 , add together, 
and divide out by 2 xy ; we obtain 
C'z 2 + C xy - A'xz - B'yz = 0 (4). 
By similar processes we obtain 
A'x 2 + Ayz - B’yx - C 'zx — 0 (5), 
B'y 2 + B zx - C'zy - A!xy = 0 (6). 
Between these six, treated as simple equations, the six 
functions of x, y, z, viz., x 2 , y 2 , z 2 , xy , xz, yz , treated as in- 
dependent of each other, may be eliminated ; the result may be 
seen, by mere inspection, to come out 
ABC(ABC - AB' 2 - BC' 2 - CA' 2 + 2A'B'C') = 0, 
or rejecting the special (hT.B. not irrelevant) factor ABC, we 
obtain 
ABC - AB' 2 - BC' 2 - CA' 2 + 2A'B'C' = 0.” (liv. 5) 
The example, • however satisfactory as illustrating the dialytic 
method, cannot he passed over without a note in regard to the 
unaccountable blunder made in developing the determinant in- 
volved. In later notation the determinant is 
• 
C 
B 
-2A' 
c 
A 
. 
- 2B' 
B 
A 
. 
- 2C' 
A' 
A 
-C' 
-B' 
B' 
-C' 
B 
-A' 
C' 
-B' 
-A' 
C 
Now neither of the factors given by Sylvester are really factors of 
this, the truth being that it 
= - 2(ABC + 2A'B'C' - BB' 2 - CC' 2 - AA' 2 ) 2 . 
