DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
595 
Section IT . — Generation of Quartic Surfaces having a Conic Nodal Line. 
Lemma. If S= < r 2 A# 2 + ;2:2 + w2 =0, A ±~ax-\- by-\-cz-\-dw= 0, then the result of the 
operation X^A dw P ei ’f° rrrie d 011 the quadric — A is 
(A a -(- igjj A cv A dfj S' — (fx a g*y A A ^c). 
which is evidently connected with S J — A by the invariant relation (13) of art 9. Hence 
we have the following theorem : — 
The result of the operation X Jr Af* A*' jz performed on a quadric of the form 
SAA. is a quadric orthogonal to S ' A A. 
25. Being given quadric J inscribed in another S and a point (a, (3, y, c>), we can find, 
by the method of the preceding lemma, a quadric Q orthogonal to J, whose pole of contact 
with S is the point (a, (3, y, l). When (a, (3, y, c>) varies, Q varies also ; and I say if the 
locus of (a, f3, y, h) is a quadric F, that the envelope of Q is a quartic surface having a 
conic nodal line. 
Demonstration. LetQ,=S* — A, Q 2 =S* — B, Q :) S- — C, Q 4 S 1 — D be four particular 
quadrics of the system cutting J orthogonally. Let us consider the quadric Q as having 
its pole of contact with S at the centre of mean distances of the poles of contact of 
Q,, Q,, Q 3 , Q 4 for any suitable system of multiples x, y, z, w, and as we are only con- 
cerned with their mutual ratios, we can put x-\-y-\-z-\-w=\. Hence we get the follow- 
ing system of eqnations : — 
a— ax +by A cz A dw, 
(3=a'x A Cy A d z A d'w, 
y = a"x A V'y A G "z A d"w, 
c5 = a!"x A l>"'y A c'"z A d'"w ; 
and from these we have Q = Qgr A Q-S A CM A Q 4 w ; and since the locus of (a, [3, y, S) is 
the quadric F, we see by substitution of the preceding values of a, (3. y, cS in its equation, 
that x , y, z, iv are connected by an equation of the second degree. If we denote this 
equation by 
(ft, h , c, ft, l, m, n, p, q, rfx, y, z, wf= 0, 
we have to find the envelope of Q^AQa AQs z AQ 4 w subject to this condition. The 
theory of envelopes gives 
a , 
n , 
m , 
Q., 
n , 
h , 
l , 
Q 2 , 
m , 
l , 
G , 
r , 
Mi 5 
V > 
r , 
d , 
Q« 
Q-4J- 
Q* 
Q», 
Q« 
o, 
4 N 
MDCCCLXXJ. 
