654] ON CERTAIN OCTIC SURFACES. 85 
the equation is 
— cx 2 + az 1 + gw 2 = 0. 
That is, k being determined by the quadric equation af+ bgk 2 + ch (k — 1) 2 = 0, the line 
PQ meets the y-conic y = 0, — cx 2 + az 2 + gw 2 = 0; and, in a similar manner, it appears 
that the line PQ also meets the ¿»-conic x — 0, cy 2 — bz 2 + fw 2 = 0. 
Writing for greater symmetry 1 : — k : k — 1 = X : y : v, we have 
X + fJb -)“ V = 0, 
af\ 2 + bgy 2 + chv 2 = 0, 
so that there are two systems of values of (X, y, v) corresponding to, and which may 
be used in place of, the two values of k respectively. 
Starting now from the equations 
(fd 2 + g) (kdy — x) 2 + h(k — l) 2 6 2 z 2 = 0, 
(bk 2 6 2 — a) (6y — x) 2 + h(k — l) 2 6 2 w 2 = 0, 
the elimination of 6 from these equations leads to an equation U = 0, of the above 
mentioned form but with a determinate value of the coefficient. 
The process, although a long one, is interesting and I give it in some detail. 
Elimination of 6 from the foregoing equations. 
We have 
U = MU [(/#- + g) (kdy — x) 2 + h(k — l) 2 6 2 z 2 ], 
where II denotes the product of the expressions corresponding to the four roots of 
the equation 
(bk 2 6 2 — a) (6y — x) 2 + h(k—l) 2 6'ho 2 = 0. 
Observing that this equation does not contain z, and that the expression under the 
sign II does not contain w, it is at once seen that the product II is in regard to 
(z, w) a rational and integral function of the form (z 2 , w 2 ) 4, ; and since, in regard to 
(z, w), TJ is also a rational and integral function of the same form (z 2 , w 2 )*, it is clear 
that the factor M does not contain z or w, but is a function of only (x, y). To 
determine it we may write z = 0, w = 0: this gives 
c 2 (bx 2 — ay 2 ) 2 (fx 2 + gy 2 ) 2 = MU. (fd 2 + g) (kdy — x) 2 , 
where 
(bk 2 d 2 — a) (dy — xf = 0, 
and the values of d are therefore ^j’ y' Hence substitutin g and 
observing that 
c 2 Ji 2 (k - l) 4 = (of + bgk 2 ) 2 ,