CIRCLES AND OF SPHERES. 



11 



All of those problems can be solved by the application of the same principle 

 which is employed in the above investigation of the tangencies of circles. 



The solutions of Problems 1 and 10 are found in all elementary works on de- 

 scriptive geometry, and are obvious. 



In Problem 2, pass a circle through the three given points, it will be a small 

 circle of the required sphere. Through its centre, pass a plane perpendicular to 

 the given plane, and also to the plane of the circle. It will cut the circumference 

 of the circle in two points, and the given plane in a line which must contain the 

 required point of contact. By Problem 2, in the Tangencies of Circles, through 

 those two points draw a circle tangent to this line, and you will have a great circle 

 of the required sphere, and its point of contact with the plane. It is obvious that 

 there are two spheres which will fulfil the conditions of the problem, as there are 

 two points of contact found by this construction. 



In Problem 3, draw a plane bisecting the angle formed by the given planes. It 

 must pass through the centre of the required sphere. Let fall a perpendicular from 

 one of the given points upon said plane, and find on it a point equidistant on the 

 opposite side. The required sphere must pass also through this last point. There- 

 fore, the problem is reduced to Problem 2. 



