68 The Three Sections, Tangencies, 



the data is four spheres, and theu make the modifications 

 necessary when we suppose one or any number of the spheres 

 to become infinitely small or infinitely great, or — in other 

 -v^ords — when we suppose one, two, three, or all of the 

 spheres to be replaced by points or planes. 



The analysis similar to my first solution is evidently as 

 follows : — 



ANALYSIS. 



Let a, b, c, d be the respective points of contact of the 

 required sphere with the given spheres A, B, C, D. 



Then the straight lines ah, ac, ad, pass respectively through 

 O, P, Q, known vertices of similitude to the pairs of spheres 

 AB, AC, AD. 



Now, if a' be any point in the surface A, and that b' , c' , d', 

 are the respective points in the sm-faces of B, C, D, made by 

 the straight lines a'O, a'P, a'Qi which are dissimilar to that 

 of a' on the surface of A, it is evident Oa' .Ob' = Oa.Ob, 

 Va'.Vc' = Va.Vc, and Qa'.Qd' = Qa.Qd, and therefore that 

 the spheres a'b'c'd', abed, have the plane OPCl as radical 

 plane. 



This being borne in mind, it is e^ddent that the tangent 

 plane to any of the given spheres at the point of contact will 

 cut the plane of section of this sphere and the sphere a'b'c'd' 

 in a straight line situated in the plane OPGi. 



But as we may assume the point a' anywhere in the surface 

 of A, we know the resulting points b'c'd', and the sphere 

 a'b'c'd', and its planes of intersection with the spheres 

 A, B, C, D, and also the intersections of these planes Avith 

 plane OPQ, and the tangent planes from these lines to the 

 given spheres, and .'. the sought sphere of contact. 



And it may be remarked that to each plane of similitude 

 OPQ there are two answerable spheres abed whose centres 

 (as also the centre of the corresponding sphere a'b'c'd) are on 

 the perpendicular from the radical centre of the four 

 given spheres to the plane OPQ, ; and, moreover, that as there 

 are eight planes of similitude OPQ there are sixteen answer- 

 able touching spheres (real or unreal in pairs). We may 

 further remark that this solution famishes a proof that the 

 twelve vertices of similitude of the four given spheres lie in 

 sixes in the eight planes OPQ, and are the vertical points of 

 a complete octahedron. 



It is also easy to see that the chords aa of the sphere A 



