DR. J. CASEY ON CYCLIDES AND SPHERO-QTJAETICS. 
601 
W=(a-5)j3 2 + («-c)y 2 +(a-d)h 2 + (a-e)s 2 =0, ( I)' 
W=(6-c)y 2 +(l>-d)Z 2 +(&-e)e 2 +(5-ay=0, ( II) 
W=(c-d)h 2 +(c-e)e 2 +(c -a)cc 2 -h(c-&)(3 2 =0, (III) • 
W ={d- e y +(d-ay+(d-b)p 2 +(d-c)>y 2 =0, (IV) 
■W^( e - a y + (e-b)p 2 + (e-cy+(e-d)l 2 =0. ( V)_ 
(34) 
If we denote the focal quadrics corresponding to these different forms of the equation of 
W by F F' F" ¥" F (iv) , we get the following as the tangential equations of the five focal 
quadrics: — 
F y a -by+(a-cy +(a-dy+(a-ey= 0,' 
F' yb-c) v 2 +(b-dy + (b-ey + (b -a)x 2 = 0 , 
F" =(c — d)f-\-(c — e)ff 2 + (c— a)x 2 +(c — b)p 2 =0, t 
¥" =(d-ey'+(d-ay + (d-by + (d-cy= 0, j 
F (iv) =(c— a)X 2 -\-(e— b)y +(<?— c)v 2 + (c — ^)g> 2 =0.j 
(35) 
So that the cyclide W can be generated in five different ways as the envelope of a vari- 
able sphere whose centre moves on a quadric and cuts a given sphere orthogonally ; 
the corresponding spheres and quadrics for generating W being a, F ; ft, F' ; y, F" ; S, ¥" ; 
2 , F (iv) respectively. 
34. Since the tangential equation of a is plainly (j 2 -{-(r 2 =Q, 
of (3, v 2 — j— — |— <r 2 — J— X 2 , See., 
we get the equations of the developables circumscribed to the pairs of quadrics «, F ; 
ft, F', &c. by a known process ; thus the developable circumscribed about a and F will 
be the envelope of the quadric, whose tangential equation 
(u — -j - (a: — c)v 2 -J- [cl — d)g~- \- ( Qj — ^)(T" -\-Jc([Jj 2 -\- v“ d - F = 0 ; 
and by taking k=(b—a), ( c—a ), ( d—a ), (e — a) in succession, we see that the double 
lines of the developable are the plane conics, whose tangential equations are : 
(i b—c)v 2 -f -(b — d)f-\-(b —e)d 2 — 0, 
(c— b)[jj 2J f(c —d)f J r (c —ey=0, 
(d-by+(d-cy + (d-ey= 0, I • * 
(e — —c)v 2j r(e — d)f= 0. 
By comparing these with the system of equations (35), we see that the first conic is a 
plane section of the quadric F', the second of F", the third of F'", and the fourth of 
Hence, if we call the spheres a, ft, y, c>, g the spheres of inversion of the cyclide (w ? e 
shall prove this in a future Chapter), and call 5, S', the five developables 
circumscribed to the spheres of inversion and their corresponding focal quadrics, we 
have the following theorem : — 
