G70 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
Hence we get, by Plucker’s equations, 
r=m{m— 1)— 2/i— 3/3, m=r(r— 1) — 2y—on, 
n = 3 m(m — 2) — GA — 8/3, (3 = 3r(r — 2) — 6y — 8 n, 
(n — j8) = 3 (r— m), 2 (y — h) = (r — m)(r + m — 9). 
221. Again, let us consider a plane section of A. Professor Cayley has shown that 
for such a section 
yj=r, v=n, b=x, r—g, z=m, i=a. 
Hence, from Plucker’s equations, we get 
n =r(r — 1 ) — 2 x — 3 m, r = n(n — l) — 2g—Zu, 
a = 3r(r — 2) — 6a-— 8m, m= 3 n(n — 2) — 8g — 8a. 
Whence also Hr. Salmon gets 
(m — a) = 3(r — , 2(.r — y) = (r — w)(r + n — 9) . 
It is plain this system might be got from the former by considering the cone whose 
vertex is any point and which stands on the cuspidal edge of 2, and then reciprocating. 
If we combine the equations of this article with those of the last, we get 
(u—(3)=2(n—m), x—y=(n—m), 2(g—h)=(n—m)(m-\-n—l), 
since Plucker’s equations enable us, being given any three singularities of a plane 
curve, to determine all the rest. The equations of this and the preceding article enable 
us, being given any three singularities of a twisted curve, to determine all the rest. In 
a succeeding article I shall point out how they may be employed to determine the sin- 
gularities of the evolute of a plane curve when three of the singularities of the original 
curve are given. 
222. Eight lines of A meet an arbitrary line. 
Demonstration. Let (LM) be the arbitrary line and P a point in (LM) where one of 
the lines of A meets it ; then it is plain, if P be the common vertex of two cones tan- 
gential to U and V respectively, one of the four common edges of the two cones will be 
a line of A, and also that the intersection of the polar planes of P with respect to U 
and V will pass through the curve UV. Now the intersection of the polar planes of 
U and V with respect to ail the points of LM is a quadric. For let P, y', z', w r , 
x ", y", z ", tv" be any two fixed points on (LM), and U', Li", V', V" be their polar planes 
with respect to U and V ; then the coordinates of any other point on (LM) will be 
lx'- \-mx", ly'-\-my", lz!-\-mz", lw'-\-mw", and therefore the polar planes will be ZU' + ??iU", 
IV' -\-mY", and, eliminating linearly, the locus required is U'V" — U"V , = 0, a quadric 
which intersects the curve UV in eight points. Hence the proposition is proved. 
223. The eight planes determined by the line LM with the eight lines of A which meet 
it are tangent planes to the reciprocal of the quadric U'V"— U"V' with respect to U. 
For since eight lines of A meet LM, eight lines of 2 meet the polar line of LM with 
