and Loci of Apollonius, &^c» 69 



pass respectively tkrougli the poles of the corresponding planes 

 OPQ in respect to sphere A^ and all through R the radical 

 centre of the four given spheres ; and therefore^ it follows, 

 that the eight lines in which the planes of intersection of the 

 eight spheres a'b'c'd' with the sphere A cut the corresponding 

 eight planes OPQ of similitude, are situated in the polar 

 plane of the radical centre R in respect to the sphere A, &c. 



Or we might determine the point a (and hence b, c, d) 

 from tlie following considerations : — Since Oa.Ob, and Pa.Pc 

 are of known magnitudes and that aO.ab has to aV.ac 

 a known ratio, therefore the point a must be on the surface 

 of a known sphere ha^dng its centre in the straight line 

 through O and P. 



Similarly, since Pa.Pc and Qa.Qid are of known magnitudes 

 and that aV.ac has to aOi.ad a known ratio, therefore the 

 point a must be on the surface of a known sphere ha^dng its 

 centre in the straight line through P and Q, Hence, as the 

 point a is on the sphere A, it follows that it must be a point 

 of intersection of the circular traces made on the sphere A by 

 the two known spheres having their respective centres in the 

 straight lines PO and PQ. 



The other solutions to Descartes' problem of the spheres, 

 which are analogous to those I have given to Apollonius' 

 problem of the circles, may be easily made : — the tangent to 

 two circles in the plane being represented by a plane touching 

 three spheres, &c. And the actual operations are very simple 

 when performed according to Mongers practical processes of 

 the geometry of figiu'ed space, known by the name of " De- 

 scriptive Geometry." 



APOLLONIAN LOCI PROBLEM. 



Given two points A, B, and the magnitudes of four lines a, 

 b, c, d, to find the locus of a point P, such that AP^ + a.b : 



