NOTE UPON A RESULT OF ELIMINATION. 
[From the Philosophical Magazine, vol. XI. (1856), pp. 378—379.] 
If the quadratic function 
(a, b, c, /, g, K$x, y, z) 2 
break up into factors, then representing one of these factors by %x + yy + %z, and taking 
any arbitrary quantities a, /3, 7, the factor in question, and therefore the quadratic 
function is reduced to zero by substituting /3£—ay, y% — a£, ay — ¡3% in the place of 
x, y, z. Write 
(a, b, c, /, g, h\(3£ — ay, 7^ - a£, ay - /3£) 2 = (a, b, c, f, g, h$a, 0, yf\ 
the coefficients of the function on the right hand are 
a = . erf + bÇ 2 
- 2M ■ ■ . 
b = c£ 2 4- . + aÇ 2 
■ - 2gig ■ , 
c = 2 + ay 2 
- i/lgy, 
>-ts 
II 
1 
«Y? 
- avi+ ̻ff + gin, 
g = • - 9V 2 
+ hnÇ- + fin, 
h = . . — hÇ 2 
+ gnK + № - o(n ; 
and it is to be remarked that we have identically 
a£ + h 77 + g£ = 0, 
hf + b?; + ff = 0, 
g£ + fi? + c'C, = 0. 
Hence of the six equations, a = 0, b = 0, c = 0, f=0, g = 0, h = 0, any three (except 
a = 0, h = 0, g = 0, or h = 0, b = 0, f = 0, or g = 0, f=0, c = 0) imply the remaining 
three.