[487 
487] 
ON THE QUARTIC SURFACES (*#£7, V, W) 2 = 0. 
5 
II 
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487] ON THE QUARTIC SURFACES (*$£7, V, Tfi) 2 = 0. 5 
this is 
For this purpose reverting to the equation 
)] - 27ay 2 w} 2 = 0, 
(x — aw cos 0) 2 + (y — bw sin 6) 2 + z 2 — k 2 w 2 = 0, 
this may be written 
2aw) (z 1 — h 2 w 2 )} = 0. 
A cos 20 + B sin 20 + G cos 0 + D sin 0 + E = 0, 
where 
A = (a 2 — b 2 ) w 2 , 
B= 0, 
y finally is 
G — — 4axw, 
D = — 4 byw, 
3 + 16a 4 w 4 } (z 2 - khv 2 ) 
E = (a 2 + b 2 ) w 2 + 2 (x 2 + y 2 + z 2 — k 2 w 2 ), 
and the equation is 
laving its centre on an 
{12 (A 2 + B 2 )- 3 (C 2 + D 2 ) + 4# 2 } 3 
- {27^. (C 2 - D 2 ) + 54BCD - [72 (A 2 + B 2 ) + 9 (G 2 + D 2 )] E + 8# 3 } 2 = 0, 
or say 
{12J. 2 - 3 (G 2 + D 2 ) + 4£ 2 } 3 - {27.4 (C 2 -D 2 ) - [72A 2 + 9 (G 2 + D 2 )] E+ 8E 3 \ 2 = 0. 
1 for the equations of 
This is of the order 12, but it is easy to see that the terms in E 6 and E i (G 2 + B 2 ) 
ds 0, b sin 0 ; hence the 
disappear from the equation, all the other terms divide by w i ; and the equation is 
thus of the order 8. 
). 
The equation may be obtained somewhat differently as follows. The equation of 
the variable sphere is 
(x — aw) 2 + (y — /3wf + z 2 — k 2 w 2 = 0, 
), 
a 2 B 2 
where (a, /3) vary subject to the condition — + p-=l. We have therefore 
3, 
aw 
x — aw — X —- = 0, 
a 2 
iciprocal surface is 
y — ¡3w — X ^ = 0, 
and thence 
a 2 x Xx 
aw = ———, x —aw — ——, 
a" -t- X a~ X 
= 0, 
+ b4x- 
F 2 + k 2 Z 2 } + F 4 = 0, 
Consequently 
x 2 t y 2 z 2 — k 2 w 2 A 
(a 2 + X) 2 + (b 2 +X) 2 + X 2 ~ ’ 
l 2 + (¿2 _ k 2 ) F 2 - k 2 Z 2 = 0. 
iciprocal surface to 12; 
to show) of the order 8. 
a 2 x 2 b 2 y 2 „ - 
(a 2 + X) 2 (b 2 + X) 2