328 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
from which we obtain 
a : b : c : / : g : h 
= 2yz (— x + 2y + 2z) 
: 2zx( lx— y + z) 
: 2xy{ 2x + 2y — z) 
: — x ( x 2 — y 2 — 2 2 + 4i/^) 
: — y (— x 2 + y 2 — z 2 + 4zx) 
: — z (—x 2 — y 2 + z 2 + 4xy), 
which are the required new forms. 
38. We have 
be — f 2 = 4x 2 yz (2x — y + z) (2x + 2y — z) — x 2 (x 2 —y 2 — z 2 + 4<yz) 2 
= (x + y + z) 2 (x 2 + y 2 + z 2 — 2 yz — 2zx — 2 xy), 
which is =0, if x + y + z = 0, or if x 2 —2x(y + z) + (y — z) 2 = 0. In the former case, viz. 
if x + y + z = 0, we find a = b = c —f = g = h = — Qxyz, and therefore 
(a, b, c, f, g, K$x, y\ z') 2 = - Qxyz (x + y' + z') 2 , 
but this corresponds merely to the value k = oo, for which the cubic is 
(x + y + z) 2 (\x + gy + vz) = 0, 
which is not a proper cuspidal curve. In the latter case, or where 
x 2 + y 2 + z 2 — 2 yz — 2 zx — 2 xy = 0, 
or, what is the same thing, *Jx-\-\ly-\-\/z= 0, we have a proper cuspidal curve. 
Article Nos. 39 to 43, relating to the Triangle of the Critic Centres. 
39. The equation 
MG , fiy , 
01 -j- A. 0i 4* /A t?i + v 
is satisfied by substituting therein 
111 
1 1 1 
x ■ y ■ z e,+x ' e,+n '■ e 2 +v 
x : y : z = 
, or X \ y \ z = -i 
0 3 + A 0 3 -\- fi 0 3 + v 
in fact, for the first set of values the equation becomes