100 
ON A CERTAIN SEXTIC SURFACE &C. 
[398 
Hence we have 
and thence 
x + y + z = Q (a A GH + bBHF + cCFG) 
= abcMQ (a + /3 + yf, 
w = abcMFGH, 
x+y+z+w= abcM {Q (a + /3 + y) 2 + FGH} 
Moreover 
and 
^abcMABC. 
xyzw = (abcf ABC (FGH) 3 Q 3 M, 
27a 2 b 2 c 2 xyzw (x + y + z + wf = 27 (abcf (ABCFGHMQf (*). 
Again 
( a 2 x + b 2 y + c 2 z) io = abcFGHMQ (a 3 A GH + b 3 BHF + c 3 CFG) 
= (abcf FGHMQ {Q(a + /3+ yf + 3FGH} 
a 2 yz + b 2 zx + c 2 xy = abcFGHQ 2 (aBCF+ bCAG + cABH) 
= (abcf FGHMQ . 2Q (a + ¡3 + yf, 
and thence 
a 2 (xw + yz) + b 2 (yio + zx) + c 2 (xy + zw) 
= 3 (abcf FGHMQ {Q(a + /3 + yf + FGH] 
= 3 (abcf A BCFGHMQ (*), 
and the two equations marked (*) verify the equation of the surface. 
To verify the derived equations, write for a moment P = a 2 (yz + xw) + b 2 (zx + yw) 
+ c 2 (xy + zw), so that the equation of the surface is P 3 — 27a 2 b 2 c 2 xyzw (x + y + z + wf = 0, 
and the derived equation with respect to x is 
3_ dP _ 1 2 
P dx x x + y + z + w ’ 
or substituting for P and x + y + z + w their values, this is 
№_(atoyABOrOSMQ + iaUQFOHi 
and similarly for y, z, and w. In particular, considering the derived equation in respect 
to w, this is 
a 2 x + b 2 y + c 2 z = abcABCQ + 2 abcQFGH 
= abcQ (ABC+2FGH), 
and we have as before 
a 2 x + b 2 y + c 2 z = Q (a 3 A GH + b 3 BHF + c 3 CFG) 
= abcQ {a + /3 A yf + 3FGH} 
= abcQ (ABC+2FGH), 
which is thus verified; the verification of the derived equations for y, z, w can be 
effected, but not quite so easily. 
The existence of the excubo-quartic nodal curve is thus established.