1 8 GEOMETRY OF MOTION. [32. 



III. Spherical Motion. 



32. The motion of a spherical figure of invariable form on its 

 sphere presents a close analogy to plane motion ; in fact, plane 

 motion is but a special case of spherical motion, since a 

 plane may be regarded as a sphere of infinite radius. 



33. By a generalization similar to that of Art. 30, the study 

 of the motion of a spherical figure on its sphere leads directly to 

 the laws of motion of a rigid body having one fixed pgint. For 

 the motion of such a body is evidently determined by the spheri- 

 cal motion on any sphere described about the fixed point. 



34. Let us consider any two positions F Q and F 1 of a spheri- 

 cal figure Fon its sphere, and let O be the centre of the sphere. 

 Just as in the case of plane motion (Art. 18) the displacement 

 F Q F 1 can always be brought about by a single rotation about a 

 point C on the sphere, or what amounts to the same, by a single 

 rotation about the axis OC. The proof is strictly analogous 



to that given in Art. 18. We 

 fi rst remark that the position of 

 the figure on the sphere is fully 

 determined by the position of 

 two of its points, say A and B 

 (Fig. n), since any third point 

 forms with these an invariable 

 spherical triangle. Let A , B^ 

 be the positions of A, B in F Q ; 

 A v B^ their positions in F l ; 

 and draw the great circles A A l 

 and B^B V Their perpendicular 

 bisectors intersect in two points C, D which are the ends of a 

 diameter of the sphere. CD is the axis of the displacement 

 F Q F l} and the angle A^CA^ or B^CB^ gives the angle of the 

 displacement. 



