NOTE ON THE TRANSFORMATION OF TWO SIMULTANEOUS 
EQUATIONS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. XI. (1871), pp. 266, 267.] 
Writing in Mr Walton’s equations (1) and (2) 
a b c a /3 y 
S’ d } d’ 8’ 8* 8 
instead of a, b, c, a, ¡3, y respectively; and putting for shortness 
A = by—c/3, F — a,8 — da, 
B = ca — ay, G —b8 — d/3, 
C = a/3 — bu, H = c8 — dy, 
the equations become 
a (b — c) b (c — a) c(a — b) _ 
F + G + ’ 
« (0 - 7) , 0 (7 “ a ) , 7 (a - 0) __ 0 
r~ + 0 ii 
Multiplying by FGH and effecting some obvious transformations, the equations become 
whence also 
aAF+ bBG+cCH = 0] 
(18) 
aAF+ßBG + yCH = 0j 
AF 2 + BG*+ CH* — 0 
(19)- 
Now regarding (a, /3, 7, 0) as the coordinates of a point in space, the equations 
(18) and (19) represent each of them a cone having for vertex the point a : /3 : 7 : 8 
= a : b : c : d, viz. (18) is a quadric cone, (19) a cubic cone; they intersect therefore 
in six lines; and it may 
be shown that these 
are 
the line 
a : ß 
: y = a 
: b : 
: c 
(twice) 2 
0 : 7 : 
8 = b 
: c : 
: d 
1 
}) 
7 : a : 
8 — c 
: a : 
: d 
1 
?> 
a : 0 : 
: 8 = a 
: b : 
; d 
1 
» 0-1' 
7 — a : a — ß 
: 8 = b 
— c : 
c — a : 
: a — b : d 1 
6, 
agreeing with Mr Walton’s result.