12 THE TANGENCIES OF 



In like manner, Problem 4 can be reduced to Problem 2. 



In Problem 5, pass a circle through the three given points. This must be a 

 small circle of the required sphere. Let fall, from the centre of the given sphere, 

 a perpendicular to the plane of this circle. Puss, through this perpendicular and 

 the centre of the small circle, a plane. This will cut the circumference of the 

 small circle in two points (found readily in construction by joining the foot of the 

 perpendicular with the centre of that circle), and it will cut the given sphere in a 

 great circle. Then (by Prob. 4, in Tangencies of Circles) pass through these two 

 points a circle tangent to the great circle. The point of contact will be the point 

 of contact of the required sphere. Two such tangent circles can be drawn, and 

 thus two spheres can be obtained to answer the conditions, viz : one convex and 

 one concave towards the given sphere. 



Problem 6. To draio a sphere through two points tangent to a given sphere.and to a 

 given plane. — Suppose the required point of contact (on the plane) to be known. 

 Through this point, and the centre of the given sphere, pass a plane perpendicu- 

 lar to the given plane, and suppose, for the sake of analysis, that to be the 

 plane of the paper. In Fig. 1, Plate VIII, let A be the given sphere referred 

 to two planes of projection (according to the principles of descriptive geometry), 

 A' X' being the ground line, let the vertical plane of projection be the given 

 plane, and (P, P) ( Q, Q) the projections of the given points. Let O and O' be 

 the "similar points," used as in the analogous problem (Prob. 5) in Tangencies 

 of Circles. Suppose, for the purpose of analysis, that Fis the required point of 

 contact on the given sphere and in the horizontal plane, and that X Y is the 

 required tangent sphere. Join Fwith O and J'. O Y X will be one continuous 

 straight line; see Prob. 5, in Tangencies of Circles. Also, O Yx O X' = O A' 

 x O O. Join one of the given points (P, P) with O. Let the point whose pro- 

 jections are MM' be the point in which this line pierces the tangent sphere. As 

 this line and O Y X' are in the same plane, cutting a small circle out of the tan- 

 gent sphere, we have O M" x O P" — Y x O X' = O A' x O O'. Thus, the 

 point whose projections are M, M' will be found by finding O M" a fourth propor- 

 tional to O P", O A', and O O', all these being known distances. We have thus 

 three points (P, P), (Q, Q), and (M, M') in the required sphere, and the question 

 is reduced to Prob. 5, Tangencies of Spheres. By this process, two solutions are 

 obtained. By the use of the point C, two more will be obtained ; making four in 

 all. Two of the required spheres will be convex, and two concave to the given 

 sphere. 



In Problem 7, pass a plane bisecting the angle formed by the two given planes. 

 Through the given point, let fall a perpendicular to this plane, and find a point 

 on the opposite side equidistant from the plane. This point must also be found on 

 the required tangent sphere. Then, through these two points, draw a sphere tan- 

 gent to the given sphere and to one of the given planes; it will be tangent, also, 

 to the other plane. This reduces the problem to Prob. 6. 



