80, 81] 



BODY WITH FIXED POINT. 



253 



Fig. 71. 



Let 0, Fig. 71, be the fixed point, and let OI be the instan- 

 taneous axis at a given instant. During the time dt suppose a 

 line 0/ 2 moves to the position OJ 2 r , and 

 during the next interval /It let the body 

 turn about this line as instantaneous axis. 

 During this interval let another line OI 3 move 

 to OJ 3 ' which then becomes the instantaneous 

 axis, and so on. We have thus obtained two 

 pyramids, one Ol^OLOI^ . . . fixed in space, 

 the other O^OZ/OZ^. . . fixed in the body, 

 and we may evidently describe the motion 

 by saying that one pyramid rolls upon the 

 other. As we pass to the limit, making 4t 

 infinitely small, the pyramids evidently become 

 cones, and the generator of tangency is the 

 instantaneous axis at any instant. 



The rolling cone may be external or internal to the fixed one. 

 In the former case, Fig. 72 a, the instantaneous axis moves around 

 the fixed cone in the same direction 

 in which the body rotates, and the 

 motion is said to be progressive, in 

 the second case, Fig. 72 c it goes in 

 the opposite direction, and the 

 motion is said to be regressive or 

 retograde. It is to be noticed that 

 it makes no difference whether the 

 rolling cone is convex (Fig. 72 a) or 

 concave (Fig. 72 b) toward the fixed 

 cone. (In the figures, in which 

 merely for convenience the cones 

 are shown circular, C denotes the 

 fixed, C 2 the rolling cone.) 



If one of the cones closes up to a line, upon which the other 

 rolls, it always remains in contact with the same generator, that is, 

 the instantaneous axis does not move. Accordingly if either cone 

 degenerates to a line, the other does also, and the instantaneous axis 

 remains fixed in space and in the body. This case has been already 

 treated. 



If we lay a plane perpendicular to the instantaneous axis at a 

 distance E from 0, Fig. 73, and if the radii of curvature of its inter- 

 sections with the fixed and rolling cones be Q and Q 2 (taken with 

 the same sign if they lie on the same side of the common tangent), 

 and the angles made by consecutive tangents at the ends of correspond- 

 ing arcs ds 1 and ds 2 are dr i9 dr if we have ds 1 



Fig. 72 a. 



