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,