93] 



NUTATION. CYCLOIDAL PATH. 



287 



the vectors whose magnitudes are b and ft this supposition is equi- 

 valent to saying that the angular momentum makes a small angle 

 with the axis of figure , as we see from 

 Fig. 100, in which the distance DE=p- bz . 

 Making this supposition, the last term in 

 116) is negligible, also that in 119). Thus 

 we obtain from 116), 



J z o 



n 



and since is supposed to be large we may 



neglect 

 122) 



123) 



I, so that we have finally, 



Fig. 100. 



s* 



pi T~ ^smbt, 



124) | = J^---*- ^si 



Vi-^ 2 T/i-V 



yr- 



It is evident from Fig. 100 that /3 &# 2 is positive, accordingly 

 [cf. 119)] the apex is always moving so that -~ is positive at the 



bottom of its path, and thus the average motion is in that sense. 

 The motion at the top may be in either direction, according to the 

 magnitude of c. We see that the motion of nutation is opposite to 

 the motion of the clock -hands. Thus the motion of the apex, as 



given by 124), is that of a point at a distance from the 



yi-V 

 center of a circle which rolls on a line above it with its center 



advancing at a velocity - - The radius of the rolling circle is 



- ~ V 



Such a locus is called a cycloid. In the ordinary cycloid, the 

 tracing point is on the circumference of the rolling circle, or 

 /3 &# = be. If the tracing point is an internal one, the cycloid 



is called prolate. It has no loops, nor vertical tangents, and -^ is 



never zero, but it has points of inflexion. If the point is external 

 the cycloid is called curtate, and has loops, but no inflexions. It is 

 evident that this curve will be described when the apex is given a 

 push to the left at the top of its motion, while if it be given a 

 push to the right it describes the prolate cycloid, and if it be simply 

 let go, it describes the ordinary cycloid with cusps. (The prolate 

 and curtate cycloids are also called trochoids.) Since the height of 

 a cycloid is to the length of its base as 1 : x, the base being the 



