[487 
487] 
ON THE QUARTIC SURFACES (*$Z7, V, Wf = 0. 
3 
; wy=o. 
ics, vol. xi. (1871), 
Le reciprocals of several 
the coordinates (X, Y, Z, W) and (x, y, z, w) is then Xx + Yy + Zz + Ww = 0, and the 
equation (X, Y, Z, W) 4 = 0 is the equation in point-coordinates of the quartic surface, 
or in line-coordinates of the reciprocal surface : and similarly the equation (x, y, z, w) n = 0 
is the equation in point-coordinates of the reciprocal surface, or in line-coordinates of 
the quartic surface. 
Parabolic ring, or envelope of a sphere of constant radius having its centre on a 
parabola. 
Taking k for the radius of the sphere, and z = 0, y 2 = 4ax for the equations of 
the parabola, then the coordinates of a point on the parabola are ad 2 , 2a6, 0; where 
6 is a variable parameter. The equation of the sphere therefore is 
(x — a6 2 w) 2 + (y — 2 aduif + z 2 — k 2 w 2 — 0, 
and the ring is the envelope of this sphere. 
The reciprocal of the sphere is 
k 2 (.X 2 +Y 2 + Z 2 ) - (ad 2 X + 2adY+ W) 2 = 0 ; 
writing this in the form 
ad 2 X + 2a0Y+ W+ k f(X 2 + Y 2 + Z 2 ) = 0, 
and taking the envelope in regard to 6, we have 
X{W+k V(X 2 + Y 2 + Z 2 )} — aY 2 = 0, 
or, what is the same thing, 
(aY 2 -X W) 2 - k 2 X 2 (X 2 + Y 2 + Z 2 ) = 0, 
soid at right angles to 
y detail, but merely for 
and with the present 
but for thfe metrical 
3qual to unity: I have 
quations shall be homo- 
e without difficulty, and 
i change. 
a, point on the quartic 
are to be considered as 
>r the coordinates of a 
2. The reciprocation is 
): the relation between 
for the equation of the quartic surface. This has the line X = 0, Y= 0 for a tacnodal 
line, but I am not in possession of a theory enabling me thence to infer that the 
parabolic ring is of the order 6. 
To show that it is so, I revert to the equation of the variable sphere 
(x — ad 2 u>) 2 + (y — 2a6w) 2 + z 2 — k 2 w 2 = 0, 
or, what is the same thing, 
(A, B, G, D, E\6, 1) 4 = 0, 
where 
A = Sa 2 w 2 , 
B = 0, 
G = a (2aw 2 — xw), 
D = — 2ayw, 
E— 3 {x 2 -f y 2 + z 2 — k 2 w 2 ). 
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