4 ON THE QUARTIC SURFACES (*$£7, V, F) 2 = 0. [487
Then I = 3a 2 w 2 I', J = a 3 w 3 J', and the equation is I' 3 — J' 2 = 0, viz. this is
{4a? 2 + 3 y 2 — 4saxw + 4a 2 w 2 + 3 (z 2 — k 2 w 2 )) 3
— {(2aw — x) [8a? 2 + 9y 2 + 4<axw — 4a 2 w 2 + 9 {z 2 — k 2 w 2 )] — 27ay 2 w} 2 = 0,
or, as this may also be written,
{4a? 2 + 3y 3 — iaxw + 4a 2 w 2 + 3 (z 2 — k 2 w 2 )) 3
— {— 8a? 3 — 9xy 2 + 12aa? 2 w + 12a 2 a?w 2 — 8a?w 3 — 9 (a? — 2aw) (z 2 — h 2 w 2 )} — 0.
Developing, the whole divides by 27, and the equation of the ring finally is
(y 2 — 4aa?w) 2 [y 2 + (a? — aw) 2 )
+ {3?/ 4 + y 2 (2a? 2 — 2axw + 20a 2 w 2 ) + 8ax 3 w + 8a 2 x 2 w 2 — 32a 3 a?w 3 + 16a 4 w 4 } (z 2 — k 2 w 2 )
+ (3y 2 4- a? 2 + 8axw — 8a 2 w 2 ) (z 1 — k 2 w 2 ) 2
+ (z 2 — k 2 w 2 ) 3 = 0.
Elliptic ring, or envelope of a sphere of constant radius having its centre on an
ellipse.
Taking k for the radius of the sphere, and z = 0, ^ = 1 for the equations of
the ellipse, the coordinates of a point on the ellipse are a cos 6, h sin 6; hence the
equation of the variable sphere is
(a? — aw cos 6) 2 + (y — bw sin 0) 2 + z 2 — k 2 w 2 = 0.
The reciprocal of this is
k 2 (X 2 +Y 2 + Z 2 ) - (aX cos 6 + b Y sin $ + F) 2 = 0,
viz. writing this under the form
aX cos 6 + b Y sin 6 + W + k f{X 2 + Y 2 + Z 2 ) = 0,
and taking the envelope in regard to 6, the equation of the reciprocal surface is
a 2 X 2 +b 2 Y 2 ={W + k f(X 2 +Y 2 + Z 2 )) 2 ,
viz. this is
(a 2 - k 2 ) X 2 + (b 2 - k 2 ) Y 2 - k 2 Z 2 - W 2 = 2k W f(X 2 +Y 2 + Z 2 ),
or
{(a 2 - k 2 ) X 2 + (b 2 - k 2 ) Y 2 - k 2 Z 2 - W 2 ) 2 - 4& 2 F 2 (X 2 + Y 2 + Z 2 ) = 0,
that is
{(a 2 - k 2 ) X 2 + (6 2 - k 2 ) Y 2 - k 2 Z 2 ) 2 - 2 F 2 {(a 2 + k 2 ) X 2 + (b 2 + k 2 ) Y 2 + k 2 Z 2 } + F 4 = 0,
which is a quartic surface having the nodal conic F = 0, (a 2 — k 2 ) X 2 + (b 2 — k 2 ) F 2 — k 2 Z 2 — 0.
This singularity alone would only reduce the order of the reciprocal surface to 12;
the reciprocal surface or elliptic ring is in fact (as I proceed to show) of the order 8.
