DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
683 
through the three single foci of the curve ; and we will now show that there are two 
great circles of double contact corresponding to each of the three small circles of inver- 
sion, J, J', J". For let F be the focal circle corresponding to J, then the great circle 
whose pole is the centre of J will intersect F in two points ; these will be the poles of 
two great circles, each having double contact with the sphero-Cartesian. Flence a 
sph ero - Cartesian has eight great circles of double contact. 
Cor. It is evident that a similar property holds for a sphero-guartic. 
252. If in the equation of art. 250 
1 cos c$4 -gr sin 2 r=(2/r-j-g' cos c$) cos § — g sin <$ sin § cos 3 
we substitute the spherical coordinates f h' of any point (see art. 36), we get a quadratic 
for yj, showing that through any point f S' two generating circles pass. Hence, reasoning 
as in the last article, through any point may he described eight circles each having double 
contact with the sphero-Cartesian. Hence, if we invert the sphere into a plane, the inverse 
of the sphero-Cartesian will not he a Cartesian oval hut a bicircular guar tic. 
253. The following properties of sphero-Cartesians are the analogues of properties of 
plane Cartesians which have appeared in the ‘ Educational Times ’ : — 
1°. Being given two small circles such that a spherical triangle can he inscribed in one 
and circumscribed to the other , the envelope of the small circle 'which has the spherical 
triangle as a self-conjugate , or , as it may more appropriately he called, cm harmonic triangle, 
is a sphero-Cartesian. 
2°. Through any point on a sphero-conic can he described three circles which osculate 
the sphero-conic ; the envelope of the circle through the three points of osculation is a 
sphero-guartic. 
3°. If a sphero-guartic with a double point O he cut by a circle in four points A, B, C, D, 
and if OK, OB, OC, OD cut the circle again in E, F, G, H, any pair of great circles 
through these points will he egually inclined to the bisectors of the angles between the 
tangents at O. 
4°. If a sphero-conic he turned through 90 J round tie principal axis of the cone which 
cuts the sphere in the sphero-conic, the locus of the intersection of any tangent with the 
same tangent in its new position is a sphero-guartic. 
5°. The locus of one set of foci of all the conics which have double contact with a 
given circle at given points is another circle passing through those points and through 
the centre of the given circle. Hence, by inversion, the locus of one set of foci of sph ero- 
guartics with a double point which have a given generating circle, and which have given 
points of contact with it, is a circle through the points of contact. 
6°. The three points in which a circular cubic is cut by any transversal are the foci 
t)f a Cartesian oval passing through four concyclic foci of the cubic. Hence, by inversion, 
four concyclic points on a sphero-guartic A are the foci of another sphero-guartic B 
passing through four concyclic foci of A. It is evident that this property is analogous 
to that of pole and polar, and that a similar use may be made of it. 
mdccclxxi. 5 A 
