398] AND SEXTIC SURFACE CONNECTED THEREWITH. 99 
and it has also a cuspidal conic, the intersection of the plane x+y- s rz + w = 0 with 
the quadric surface 
a 2 (xw + yz) + b 2 (yw + zx) + c 2 (zw + yz) = 0; 
it may be observed that the four conics of osculation are also sections of this surface. 
The surface has also a nodal curve, the equations of which might be obtained by 
inversion of those of the nodal curve of the sextic torse above referred to; but I 
prefer to obtain them independently, in a synthetical manner, as follows: 
Take a, ft, y arbitrary, and write 
— A = (b — c) a 4- bft — cy, F — by — eft, 
— B — (c — a) ft + cy — act, G = col — ay, 
— G = (b — c) y + act. — bft, H — aft — ba, 
M = (b — c) a. + (c — a) ft + (a — b) y, 
Q = a? (b — c) cl + 6 2 (c — a) ft + c 2 (a — b) y: 
then it is to be shown, that not only the equation of the surface is satisfied, but that 
also each of the derived equations is satisfied, by the values 
x : y : z : w = aAGHQ : bBHFQ : cOFGQ : abcFGHM; 
each of the quantities A, B, C, M, Q is linearly expressible in terms of F, G, H, 
which are themselves connected by the equation aF +bG+ cH = 0; the foregoing values 
of x, y, z, w are consequently proportional to quartic functions of a single variable 
parameter, say F+-G; and there is thus an excubo-quartic nodal curve. 
To establish the foregoing result, we have 
aA +bH + cG =0, 
aH -\- bB + cF = 0, 
aG + bF + cC = 0, 
aA + bB + cC = 0, 
aF + bG +cH = 0, 
F+ G + H =-M, 
bcF + caG + abH = Q, 
2bcF = a 2 A — b' 2 B — c 2 G, 
2 caG = — a? A + b‘ 2 B — c 2 G, 
2abH=-a?A -b 2 B + c 2 C, 
aAGH +bHBF +cFG = abcM (a + ft + y) 2 , 
aGH +bHF + cFG =-abc(* + ft + y)\ 
a 3 AGH + b 3 HBF + c 3 GFG = - abc {Q (a + ft + y) 2 + 3FGH), 
aBGF + bCAG + cABH = ZMabc (a + ft + y) 2 , 
Q (a + ft + y) 2 + FGH = A BG, 
which are all of them identical equations; but as to some of them the verification is 
rather complex. 
13—2