76 
ON THE CUBIC CENTRES OF A LINE &C. 
[315 
which is identically true in virtue of a + b + c = 0 (in fact, multiplying out, this gives 
12a 2 6 2 c 2 + 4 (b 3 c 3 + c 3 a 3 + a 3 b 3 ) + abc (a 3 + b 3 + c 3 ) 
— 8a 2 6 2 c 2 — 4 (b 3 c 3 + c 3 a 3 + a 3 b 3 ) — 2abc (a 3 + b 3 + c 3 ) — a 2 b 2 c 2 = 0; 
that is 
3a 2 6 2 c 2 — abc (a 3 + b 3 + c 3 ) = 0, or abc (a 3 + b 3 + c 3 — Sabc) = 0, 
where the second factor divides by a+b + c), we find the above-mentioned equation, 
We then have 
x+y+z+ 3 P = 0. 
—x+y+z _ x + y + z 2x _ 6a 2 _ 3 be 
P P P + 2 a 2 + be 2a 2 + be ’ 
that is 
— x + y + z _ — 3be x — y + z _ — Sea x + y — z _ — Sab 
P 2a 3 + bc > P 2b 2 + c ’ P ~ 2c 2 + ab ’ 
and forming the product of these functions, and that of the foregoing values of 
p, p, p, we find as before, 
— (—x + y + z){x — y + z) (x + y — z) + xyz = 0 
for the equation of the locus of the single centre. The equation shows that the locus 
is a cubic curve which touches the lines x = 0, y = 0, z = 0 at the points where these 
lines are intersected by the lines, y — z = 0, z — x = 0, x — y — 0 (that is, it touches the 
lines x = 0, y = 0, z = 0 harmonically in respect to the line x + y + z = 0), and besides 
meets the same lines x = 0, y = 0, z = 0 at the points in which they are respectively 
met by the line x + y + z = 0. 
2, Stone Buildings, W.C., September 25, 1861.