DR. J. CASEY ON CYCLIDES AND SPIIEEO-QUAETICS. 
679 
The binodal cyclide (a, b, c,f, g, hja, jS, y) 3 =0 will be intersected by the sphere U 
orthogonal to a, (3, y, and whose centre is complanar with their centres in a sphero- Carte- 
sian if the conic ( a , b , c, f, g, Jiff, v) 2 pass through the centre of U. 
241. From the method of generating spliero-Cartesians given in art. 239, Cor. 2, we 
can get one form of its equation considered as a curve described on a sphere. 
Thus, let the equation of the sphere of which, the circle J 
is a great circle be x 2 -\-y 2 -\-z 2 — 1, and the equation of F be 
{y + hf 2 = 4a(h + x), or, in polar coordinates, 
(g sin 0-j-h') 2 =4 a{h-Pq cos 6), 
and it is clear that the perpendicular to the plane of the 
paper at P will cut the sphere in a point Q whose spherical 
coordinates are, thus determined. 
Taking the great circle J as a circle of reference, making 
AP:=0, PQ perpendicular to it =\J/, then we have cos \f/ = §, 
and the equation required is 
(sin d cos \|/+^) 2 =:4(2(7i+ cos 0 cos \f/). . . . (149) 
242. Let the great circle J of the sphere intersect the parabolic base of the cylinder in 
four points, and let K, K', K" be the points of intersection of the three pairs of lines 
through these four points, the sides of the triangle K K' K" will cut off from the sphere 
three arcs, and the three small circles which have these three arcs as spherical diameters 
will be the three circles of inversion of the sphero-Cartesian. Again, the three pairs of 
perpendiculars from the centre of the sphere on the three pairs of opposite connectors 
will cut the sphere in three pairs of points which will be the extremities of the diameters 
of the three focal circles of the sphero-Cartesian. lienee, being given on the surface of 
a sphere U a focal circle F and a circle of inversion J of a sphero-Cartesian, we infer 
the following construction for the two remaining focal circles and circles of inversion : — 
Let H and O be the centres of J and F, and let HO intersect the circles J and F in 
the points A, B, C, I) ; if the points P and Q be taken so as to be common points of 
harmonic section of A B and CD, then P and Q are plainly the points in which radii 
from the centre of U to the points K', K" pierce U ; they are therefore the centres of inver- 
sion of the sphero-Cartesian. 
243. Again (see tig. 5, art. 198), if the circles F', F" be described on the sphere as 
they are in the diagram referred to on the plane, we shall have the three focal circles 
and their radii given by the following equations: — 
tan 2 r — tan PFI . tan QH, j 
tan 2 r' — tan OH . tan QH, l (150) 
tan 2 r"— tan OH . tan PH . J 
The first equation is evident, since P, Q are conjugate points with respect to J ; and 
Fig. 6. 
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