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DK. J. CASEY ON CYCLIDES AND SPHEEO-QUAKTICS. 
where 
cos 2 6 -f- cos 2 <p -f- cos 2 = 1, 
A 2 . 
and then inverting by putting §=—, and changing back again to Cartesian coordinates, 
we evidently get an equation of the same form, which proves that the inverse of a 
cyclide is a cyclide. 
74. If the absolute term in the equation (75) vanishes, it is evident that the coefficient 
of (a ,2 H-v/ 2 +c 2 ) 2 in the inverse surface vanishes; in other words, the inverse surface will 
be a cubic cyclide, that is, a cubic surface passing through the imaginary circle at infi- 
nity. The section of this cyclide by any plane will be a circular cubic, and its focal 
quadric will be a paraboloid. Hence the inverse of a cyclide from any point on the 
cyclide will be a cubic cyclide. This corresponds to the theorem that the inverse of a 
bicircular quartic from any point on the quartic itself will be a circular cubic. 
75. If 5=0 in the general equation (75), that coefficient vanishes in the inverse 
surface, that is, the inverse surface will want the absolute term. Hence if we invert a 
cubic cyclide from any point not on the cubic itself, the inverse surface will be a quartic 
cyclide passing through the origin. 
76. If in the general equation (75) not only s vanish, but also the coefficients of x,y, z 
in I!) each separately vanish, that equation will represent a quadric, and the centre we 
invert from will be a node on the inverse surface. Hence the inverse of a quadric will 
be a cyclide having the origin or centre of inversion as a node. 
The species of the node will depend on that of the quadric which is inverted. 
1°. If the quadric inverted be central, let its equation be 
(x-af , {y-hf [z-cf 
L M ‘ N ~ x — u - 
If this be inverted by the process of art. 73, we get 
i\f^ i „, 2 L ^2 owJHa _ b ljL. c l 
+H+H- 1 )^+^+^- M5 (f+S+l)^+^+^+^(T+H+ 2 N)= 0 - (™) 
X 2 qp P 
Hence the node has the cone p + + 
CC^ 'IJ*' z ~ 
+ \I +i^/ or a tangent cone -> that is, the node is a conic 
node ; the cone will evidently be real or imaginary according as the quadric inverted is an 
hyperboloid or ellipsoid. 
2°. Let the quadric inverted be non-central ; let its equation be (see Salmon’s ‘ Geo- 
metry of Three Dimensions ’) 
ax 2 +by 2J r 2 px + 2 qy -j- 2rz -f- d= 0. 
The process of art. 73 gives for the inverse surface 
hi (ax' 2 + by 1 ) + 2£ 2 ( px-\-qy -j- rz) {x 2 J r y 2 + z 2 ) + d(x * + y 2 + z 2 ) 2 = 0 . . . (77) 
Hence the tangent cone at the node reduces to the pair of planes ax 2 +by 2 —0, and the 
pair of planes will be imaginary or real according as the quadric inverted represents an 
elliptic or a hyperbolic paraboloid. 
