398] 
AND SEXTIC SURFACE CONNECTED THEREWITH. 
89 
It may be mentioned that we have identically 
ae — 4bd + 3c 2 = 0, 
af — 3be -f- 2cd = 0, 
ag - 9ce + 8d 2 = 0, 
bg - 3cf + 2de = 0, 
bf — 4ce + 3d 2 = 0, 
and moreover 
ag — 6bf + loce — 10d 2 
= -6(bf- 4ce + 3d 2 ), = + 6(/ 3 -27/ 2 ), 
so that the equation of the developable may be written in the form 
or in the more simple form 
ag — 6bf + 15ce - 10d 2 = 0, 
bf — 4ee + 3d 2 = 0, 
each of which puts in evidence the nodal curve on the surface. 
The nodal and cuspidal curves meet in the points 
a bed 
b c d e ’ 
being, as it is easy to show, a system of four points. The four points in question 
form a tetrahedron, the equations of the faces of which may be taken to be x = 0, 
y = 0, z = 0, w = 0; and the equation of the surface may be expressed in this system 
of quadriplanar coordinates. 
We introduce these coordinates ab initio, by taking the quartic function of t to be 
(a, b, c, d, e$t, l) 4 = x(t + a) 4 + y {t + /3) 4 + z (t + y) 4 + w(t + 8)\ 
that is, by writing 
a — x-\- y + z + w, 
b = ax + /3y + yz + 8w, 
c = o?x + (3 2 y + 7 2 z + B' 2 w, 
d = a?x + ¡3 s y + y 3 z + B 3 w, 
e = a*x -f /3 4 <y + 7 A z + 8%. 
Observe that (ti, t 2 , t 3 , i 4 ) being any constant quantities, we thence have 
e - dSij + cltj, - UZtMi + atdMi 
= x (a - ¿0 (a - 4) (a - 4) ( a - U) 
+y(/3-t 1 ) {P-QW-QW-Q 
+ z (y - ¿i) (7 - Q (7 — Q (7 — Q 
+ w(B — i x ) (8 —1 2 ) (8 —1 3 ) (8 — ¿4), 
C. VI. 
12