CIRCLES AND OF SPHERES. 13 



In Problem 8, pass three auxiliary planes parallel to the three given planes, and 

 distant from each the radius of the given sphere. Through its centre, draw (by 

 Prob. 4) a sphere tangent to these three auxiliary planes. Its centre will be the 

 centre of the required sphere. Four solutions will be found ; two of the tangent 

 spheres concave, and two convex to the given sphere. 



Problem 9. To draw a sphere through iwo points, and tangent to two given spheres. 

 — Let A and B (see Fig. 2, Plate VIII) be the two given spheres, and G and D 

 the two given points; not necessarily in the plane of the paper. Join Cwith the 

 point O, the " external similar point' of the two given spheres. By a method 

 analogous to that explained in Prob. 6 (in Tangencies of Circles), we can find the 

 point, C" in which the line O G pierces the required tangent sphere. For, suppose 

 F and F' to be the recpuired points of contact (not necessarily in the plane of the 

 paper). Join F F', and the line will pass through the same point O, as seen in 

 the problem just quoted. The lines F F' and G G', passing through the same 

 point O, are therefore in the same plane, cutting a small circle out of the required 

 sphere. Therefore, O C x O C = O F x O F' = O H x O G. Thus, find O C 

 a fourth proportional to O C, O G, and O H, and you have O C. This gives us 

 the point C. This reduces the problem to that of Prob. 5. By the use of the 

 point O, two of the solutions will be found. By making use of the point O, the 

 " internal similar poinf of the two given spheres, two other tangent spheres will 

 be obtained ; making in all four solutions to the problem. 



In Problem 11, join the given point with one of the similar points of the two 

 spheres, and, by the method emploj'cd in Prob. 9, find another point, which must 

 be on the required sphere. Then, by Prob. 6, through these two points, draw a 

 sphere tangent to the given plane, and to one of the given spheres ; it will also be 

 tangent to the other sphere. Eight solutions will be found to this problem. Call 

 r and r' the "similar points" of the two spheres, and o and o' of one of the spheres 

 and the plane. Two of the solutions are obtained by the use of o and r, two from 

 o and r", two from o' and r, and two from d and r'. The eight solutions are classi- 

 fied as seen in Prob. 12. 



Problem 12. Todraiv a sphere tangent to two planes ami to two spheres. — Pass two 

 auxiliary planes parallel to the two given planes, outside of them and distant from 

 each the radius of the smaller of the given spheres. Pass within the larger sphere 

 an auxiliary sphere, having the same centre, and distant from its surface by the 

 above named radius. Through the centre of the smaller sphere, draw (by Prob. 7) 

 a sphere tangent to these two auxiliary planes and to the auxiliary sphere. Its 

 centre will also be the centre of the required sphere. By passing an auxiliary 

 concentric sphere outside the larger sphere, and auxiliary planes on the opposite 

 _ side of the given planes, other tangent spheres will be found. There are eight 

 solutions in all. The classification will be analogous to that in Prob. 9 (in Tan- 

 gencies of Circles), and the same as in the preceding problem, viz : two of the re- 



