624 
DE. J. CASEY ON CYCLIDES AND SPI1EEO-QUAETICS. 
Now, reasoning as in art. 69, if P become a point, since sin 3 ^— — ? (see art. 36), we 
have the following theorem : — 
The results of substituting the spherical coordinates of any point P on the surface of a 
sphere U in the equations of two small circles on U, divided respectively by the sines of the 
radii of these circles , have a ratio which is unaltered by inversion. 
From this theorem it follows, as in art. 70, if W fa, b , c, d, l, m, n, p , q, rfv, (3 , y, ci) 2 =0, 
where a, [3, y, b are small circles on U, be the equation of any sphero-quartic, that the 
will evidently cut orthogonally the sphere inverse to the plane z—a, that is, the sphere 
x 1 + y 1 + z 2 - A(a.r + <py + az) + i 
will cut orthogonally the sphere a? 2 +y 2 -f-z 2 =h, 
a 
and its envelope will be the inverse of the before-mentioned cone of revolution. 
Let the coordinates of the centre of 
x ~ -\- y 2 + - 2 — — ( ax + g>y + az) + — be X, Y, Z, 
12 12 
hence 
a=OX, (3 = OY, a=Q,Z ■ 
ftX „ aY 
“=TT 
Hence, restoring the value of £2, we get 
cc — {(1 - m 2 )(a 2 + /3 2 ) + 2m?ca + a- - m 2 c 2 }X ; 
and substituting the values of a and j3 we get 
(1 — m 2 )a 2 (X 2 -f Y 2 ) + (a- — m 2 c 2 )Z 2 + 2 m 2 c«XZ — «Z = 0, 
or, say, y = 0 ; and therefore the quadrinodal cyclide, which is the inverse of the before-mentioned cone of revolution, 
is the envelope of a variable sphere whose centre moves on the quadric v = 0, and which cuts orthogonally the sphere 
III. The plane Z = 0 touches y = 0 at an umbilic ; and it is also a tangent plane to the sphere x 1 + y 1 J r z 2 = - • 
a 
and the centre of the sphere is on one of the principal axes of y. Hence it touches y at another umbilic; but 
if a sphere touch a quadric at two umbilics, it touches at two others. Hence we have the following theorem : — 
A quadrinodal cyclide is the envelope of a variable sphere whose centre moves on a given quadric, and which cuts 
orthogonally a sphere which touches the quadric at four umbilics. 
Cor. "We can get from the canonical form of the equation of the cyclide in terms of its five spheres of inver- 
sion the condition for four nodes. Thus, let the cyclide be 
we have 
aod + bj3 2 + cy 2 •+ clS 2 + es 2 — 0, 
a 2 + j3 2 + y 2 + S 2 + e 2 = 0 identically. 
Now if two of the coefficients a, b, c, d, e be equal to one another, for instance d and e, we get for the focal 
quadrics three quadrics and one conic, which must be a focal conic of the three focal quadrics. The cyclide 
will therefore in this case have two nodes. 
If two distinct pairs of the five quantities a, b, c, d, e be equal, such as b=c, cl = e, then we get for the focal 
quadrics one quadric and two conics, and the cyclide will have four nodes. 
IY. If — — -f- (—■ -(- — — 1 = 0 be an ellipsoid, then the sphere 
a 2 lr c - 
H ) 5 
+r+- 
2 _ (be V 
(v 
j touches it at four um- 
