700] 
CASE OF THE 16-NODAL QUARTIC SURFACE. 
439 
functions of x, y, z, w have to be multiplied in order to reduce them each to the 
corresponding linear function of X, Y, Z, W, being given by the table 
1, 1, 1, F 
1 
mft 
(la - 
■ mft), 
a'a" 
m 
(la — 
711ft), 
a a 
n 
(la 
- mft), 
W'y" - 
a a 
-ftW), 
ft"ft 
l 
(mft' - 
■ ny'), 
1 
ny 
(mft' - 
ny'), 
_ft"ft 
n 
(mft' 
~ ny), 
g(Va- 
- 7«"), 
77' 
l 
(ny" - 
- la"), 
77' 
m 
(ny" - 
la"), 
1 
la!' 
(ny" 
-la"), 
77 
- aft). 
For instance, we 
have 
a (y'y"Y— ft'ft"Z) - 
- W = 
Y la - 
- mft) (fx - gy - 
- hz), 
viz. substituting for Y, Z, W their values, the relation is 
in aft . y'y" (— nx * + lz + gtu) \ 
— maft . ft'ft" ( mx — ly * + hw) i = (la- mft) (fx — gy — hz). 
— mft (—fx — gy — hz * ) ) 
As regards the terms in y, z, and w, the identity is at once verified. As regards 
the term in x, we should have 
maft (— ny'y" — mft'ft") — (la — 2mft) f= 0, 
viz. substituting for f its value, — laa'a" = — onaa'ft", the equation divides by ma and 
we then have 
ft (- ny'y" - mft'ft") + aft" (la - 2m ft) = 0, 
that is, 
la a'ft" — mft ft" (ft' + 2a) — nfty'y" = 0, 
or writing herein mft" = la!', ny' = la!, and ft' + 2a! = a — y, the equation becomes 
a a ft" — a" ft (a — y) — a! fty" = 0, that is, a! (aft" — a"ft) = a'fty" — a'fty ; or writing herein 
a'fty' = a!ft"y, the equation divided by a' becomes aft" — a!'ft = fty" — ft"y, which is true 
in virtue of a + ft + 7 = 0 and a!' + ft" + y" = 0. And in like manner the several other 
identities may be verified. 
The equation a'ft"y = a'fty might have been obtained as the condition of the 
intersection, in a common point, of four of the singular planes of the 16-nodal 
surface; and when this equation is satisfied, there are in fact four systems each of 
four planes, such that the four planes of a system meet in a common point: viz. we 
have 
Planes 
X =0, ftyX + yaY + aftZ = 0, y'(ft"ftX-a"aY)-W=0, ft" (aa'Z - yy'X) - IF = 0, 
F = 0, 7 (ft'ft"X — a'a"Y) — W = 0, ft'y'X + y'a'Y + a'ft'Z =0, a" (yy'Y - ftft'Z) - W = 0, 
Z = 0, ft (a'a'Z - y'y"X) -W=0, a' (y"y Y - ft"ft Z) -W = 0, ft"y"X + y"a" F+ a!'ft"Z = 0, 
W = 0, a (y'y” Y- ft'ft"Z) - W = 0, ft' (a"aZ - y"yX) - W = 0, y" (ftft'X - aa!Y) -W= 0,