DR. J. CASEY ON CYCLIDES AND SPHERO-QITARTICS. 
623 
Section II. — Sphero-quartics. 
81. The theory of the centres of inversion of spliero-quartics is in a great measure 
identical with that of cyclides. Thus the analogue of the fundamental theorem (art. 69) 
is the following. If a, (3, P be any three small circles on a sphere U, r,, r 2 , r 3 their 
radii, t , t 1 the common tangents of a, P ; (3, P respectively ; then, if this system be 
inverted from any arbitrary origin on U, by the formula 
tan tan constant, (83) 
g, o' being arcs drawn from the origin to a point and its inverse, we get, as in art. 69, 
sin 2 it _ sin 2 iq sin 2 JT _ sin 2 if, 
tan 7-j ’ tan r 2 tan ' tan r' 2 ' ' 
terly Journal, vol. x. p. 34, vol. xi. p. 15. The properties of quadrinodal cyclides were first studied by Dcrix. 
The principal ones will be contained in the following propositions, considered from my point of view. 
I. If a quadric of revolution he inverted with respect to any point, the inverse surface will he a trinodal 
cyclide. 
Demonstration. Let S he the quadric, L a plane of circular section, P the centre of inversion, then the sphere 
W, which passes through P and through the circle of intersection of S and L, will intersect S in another circle 
M, and will pass through a circle having OP as radius (0 being the point in which a plane through P parallel 
to L and M is intersected hy the axis of revolution of S). Now if we invert from P, the sphere W will invert 
into a plane Y, the circles of intersection of L and M with S will invert into circles Q, E in the plane Y, and 
the fixed circle having OP as radius will invert into the radical axis of Q, and R, and the points in which Q 
and E meet this radical axis will be nodes of the cyclide into which S inverts ; the point P will be a third node. 
Hence the proposition is proved. 
Cor. 1. If the quadric of revolution he a cone, the inverse of its vertex will he a node of the cyclide. Hence 
the inverse of a cone of revolution will he a quadrinodal cyclide. 
Cor. 2. Since a cone of revolution is the envelope of a variable sphere which touches three planes, we infer 
by inversion that a quadrinodal cyclide is the envelope of a variable sphere which touches three fixed spheres. 
If in this mode of generation the points common to the three spheres be real, two of the four nodes of the 
cyclide will be real and two imaginary. 
If the points common to the three spheres be imaginary, the four nodes of the cyclide will be imaginary. In 
this case it is evident the three spheres may be inverted into three spheres whose centres are collinear ; now the 
envelope of a variable sphere which touches three spheres whose centres are collinear is evidently a ring formed 
by the revolution of a circle round an axis in its plane. Hence the inverse of such a ring is a quadrinodal cyclide 
whose four nodes are imaginary ; in fact, a ring formed by the revolution of a circle round an axis in its plane 
is a quadrinodal cyclide which has two imaginary nodes at infinity. 
II. The envelope of the sphere 
(x — a)- + (y — {3 )- + (z — af = m~ {(a — c)-+/3')}, 
which cuts orthogonally the plane z—a, will, if a, /3 vary, be a cone of revolution, that is the envelope of 
x- + y" + £• — 2(a.x -\-fiy + az ) + (1 — m")(a. 2 + /3 2 ) + 2 nvea. + a- — mV= 0 ; 
or, putting for a moment 
(1 — m 2 )(a 2 -f /3 2 ) + 2m 2 ccc -fa 2 — m 2 c 2 = 12, 
the envelope of 
aP + y°- + z 2 - 2(ax + f3y + az) + 12 = 0 
is a cone of revolution. Now the sphere inverse to 
x" + y" + z- - 2(xx + py + az) + 12 
