14 " TIIETANGENCIESOF 



quired spheres convex to the two given spheres, two concave to them, two convex 

 to one and concave to the other, and two concave to one and convex to the otlier. 



Problem 13. To draw a sphere through a given 'point, and tangent to three given 

 spheres. — Let P (see Fig. 1, Plate IX) be the given point, not in the plane of the 

 paper, and ^i, B, and C, the centres of the three given spheres. Find the points 

 0, M, and ^V, the "external similar points" of the spheres taken in pairs, and O', 

 M', and N', their "internal similar points." The three points O, M, and N, are 

 shown,' by a well-known proposition, to be in the same straight line. Join the 

 given point P with each of these points, and, by the principles explained in 

 problem 9, three other points will be found, which must lie on the required 

 sphere. These four points will be in the same plane (as O M N\s a straight line), 

 and on a small circle of the required sphere; and the solution is thus reduced to 

 Prob. 5. By making use of the points O', M', and N', other tangent spheres will 

 be found to answer the conditions of the problem. There are eight solutions in all, 

 viz : one of the tangent spheres convex to the three spheres ; one, concave to all ; 

 three, convex to one and concave to the other spheres ; three, concave to one and 

 convex to the other spheres. 



Problem 14. To draw a sphere tangent to one plane and to three spheres. — Pass an 

 auxiliary plane parallel to the given plane on the side farthest from the spheres, 

 and distant the length of the radius of the smallest of the given spheres. Within 

 the two otlier spheres, pass auxiliary concentric spheres, and distant from their sur- 

 faces by the above-named radius. Pass, by Prob. 11, through the centre of the 

 smallest of the given spheres, a sphere tangent to these two auxiliary spheres and 

 to the auxiliary plane. Its centre will be the centre also of one of the required 

 spheres. By passing auxiliary concentric spheres outside of the two largest 

 spheres, and another auxiliary plane on the other side of the given plane, other 

 tangent spheres will be found to answer the conditions. The process is analogous 

 to that pursued in Prob. 9, in the Tangencies of Circles. Sixteen solutions will be 

 found to this problem, classified as follows, calling the three given spheres A, B, 

 and G: — 



Two convex to all the spheres. 



Two concave to all. 



Two convex to A and concave to B and C. 



Two B A and O. 



Two " C " A and B. 



Two concave to A and convex to B and O. 



Two " B " A and C. 



And Two " C " A and B. 



Problem 15, and last. To draio a sphere tangent to four given spheres. — In Fig. 

 2, Plate IX, let A, B, G, and D be the centres of the given spheres. The centres 

 A, B, and G may be regarded as in the plane of the paper, D being out of it. 

 Within the three largest spheres, A, B, and C, pass auxiliary concentric spheres 

 A E, B I, and G G, distant from their surfaces by the length of the radius of the 

 smallest of the given spheres. By Prob. 13, pass through D (the centre of the 



