569] 
70 
ON THE CYCLIDE. 
[569 
If in the last-mentioned equations of the circles G, C' we write x = il cos 0, 
y = il sin 6, and put for shortness 
p = a cos 0 — V, <x = m {a cos 0 — V ), 
p = a cos 0 + V, a' = m (a cos 0 + V ), 
where V = *J(c 2 — a 2 sin 2 #), then the values of il for the first circle are p, a, and those 
for the second circle are p, cr'. Hence the equations of the generating circles are 
z 2 + (i— p ) (r — a) = 0, 
z 2 + (r + p) (r — a ) = 0, 
where r is the abscissa in the plane of the circles, measured from the point x = Q. 
Attending say to the first of these equations, to find the equation of the cyclide, we 
must eliminate 0 from the equations 
z 2 + (r — p)(r — a) = 0, x = r cos 0, y = r sin #; 
the first equation is 
2 2 + r 2 + m (a 2 — c 2 ) — r(p + a) = 0, 
and we have 
p + a =(m+ 1) a cos 0 — (m — 1) \/(c 2 — a 2 sin 2 0), 
¿Hid th6nc6 
(p + a) r = (m + 1) ax — (m — 1) V{c 2 (x 2 + y 2 ) — a 2 y 2 ), 
so that we have 
z 2 + a 2 + y 2 + m (a 2 — c 2 ) — (m +1) a# + (m — 1) \/{c 2 (¿c 2 + y 2 ) — a 2 ?/ 2 } = 0, 
viz. this is the equation of the cyclide in terms of the parameters a, c, m, the origin 
being at the point x = Q, the centre of symmetry of the circles G, O'. 
Reverting to the former origin at the centre of the cyclide, we must write x — Q 
for x; the equation thus is 
{y 2 + ¿ 2 + {pc - Q) 2 - (m + 1) a (x - Q) + m (a 2 - c 2 )} 2 - (m - l) 2 [{c 2 (x - Q) 2 + (c 2 - a 2 ) y 2 }] = 0, 
where 
whence also 
i 28 - 2y 1 .. v,(/-&)(#-^) 
m + l=^_—, m- 1= JZT g > a ~~ c = Hf-gy- -jr • 
After all reductions, the equation assumes the before-mentioned form 
(y 2 + z 2 ) 2 + 2x 2 (y 2 + z 2 ) + Gy 2 + Hz 2 + (x — /) (x — g) {x - h) (x - k) = 0. 
The equation may be written 
(x 2 + y 2 + z 2 ) 2 + (G + H + K) x 2 + Gy 2 + Hz 2 — /3<y8x + fghk — 0, 
and if we ( 
then we hi 
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