654] ON CERTAIN OCTIC SURFACES. 81 
In fact, the equation of the surface may be written in the form 
iv 4 \f~x 4 + g-y 4 + h 2 z* — 2ghy 2 z 2 — 2hfz 2 x 2 — 2fgx 2 y 2 } 
+ 2w- f— cfx 4 y- — agy 4 z- — bhz 4 x 2 + 2kx l y 1 z l 
\+ bfx A z 2 + cgy 4 x- + aha?y 2 J 
+ [ay 2 z 2 + bz 2 x 2 + cx 2 y‘ 2 } 2 = 0, 
which puts in evidence the nodal curve 
w = 0, — ay 2 z 2 — bz 2 x 2 — cx 2 y 2 = 0: 
there are three similar forms which put in evidence the other three nodal curves. 
The four curves are so related to each other that every line which meets three 
of them meets also the fourth curve; there is consequently a singly infinite series of 
lines meeting each of the four curves; these break up into four series of lines each 
forming an octic scroll, and each scroll has the four curves for nodal curves respectively; 
that is, each scroll is a surface included under the foregoing general equation, and 
derived from it by assigning a proper value to the constant k. To determine these 
values, write 
A, fi -f- v = 0, 
■no' 2 ' 9 
Ar fJi- V - 
equations which give four systems of values for the ratios (A, : fi : v). We have then 
k = af V . ^ + bg ——- + ch 
A fjb \ 
viz. k has four values corresponding to the several values of (A : g : v). 
The scroll in question is M. De La Gournerie’s scroll Si; the equation of the 
scroll Si is consequently obtained from the octic equation by writing therein the last- 
mentioned value of k. 
It is to be noticed that k is, in effect, determined by a quartic equation; and, 
that, for a certain relation between the coefficients, this equation will have a twofold 
root. Assuming that this relation is satisfied, and assigning to k its twofold value, 
the resulting scroll becomes a torse; that is, two of the four scrolls coincide together 
and degenerate into a torse; corresponding to the remaining two values of k we have 
two scrolls, companions of the torse. In order to a twofold value of k, we must have 
af _ bg_ch 
A 3 g 3 v 3 ’ 
and thence 
{aff + (bgf + (chf = 0; 
or, what is the same thing, 
C. X. 
(af+ bg + ch) 3 — 27 abcfgh = 0. 
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