22 THE PENDULUM AND GYROSCOPE 



Develop cos 0= (1 sin 2 0) 1 and we get 



*),_ CW / L , 2 != 1 sin 2 0-| sin 4 0-&c.) 

 dT 22A sin 2 



For ordinary ranges of pendulum vibration the terms involving positive powers 

 of sin 6 (which express an excess of nodal motion for large excursions) may be 

 omitted, and we have 



d4>_ _Cn Lj ,^ ! c 

 dt~ ~2A~r~A~ sm 2 



Thus we see that for small vibrations, whether spherical or plane, the azimuthal 

 motion is made up of a uniform progression of the nodes - , and a fluctuating 



term which represents the angular velocity in the orlit. Indeed we have, in the 

 second term, the motions of the spherical pendulum. 



If we suppose the pendulum to have been propelled from a state of rest in the 



vertical, must have a finite value when Q is indefinitely small, and c must hence 

 dt Q n 



be unity. Hence we see that at the very outset the initial value - c r must be attri- 

 , , 



buted to , and that the pendulum reacquires it at every return excursion, that 

 dt 



is, whenever 6 diminishes indefinitely. Hence the pendulum continues to pass 



through the vertical at every return. The horizontal pro- 

 jection of the curve would be a scries of loops radiating 

 from a common centre. For each complete vibration 



the integral of - dt would represent the entire angular 



motion of the nodal axis (much exaggerated in the dia- 

 gram) from A to A', &c,, and the integral of the remain- 

 ing terms should be 2?t. These loops are in fact but 

 the path a pencil attached to a common pendulum 

 would trace upon a paper beneath, turning with uniform 

 angular velocity about the projection of the point of suspension. 



Though the numerator of the fraction - - becomes zero for the case just con- 



A sin 2 



sidered, it is evident that, at the moment of passing through the vertical, the limiting 

 value must be considered infinity, and that the integral through the infinitely short 

 time of passage must be n; for the azimuthal position undergoes, at that instant, 

 an increment (or decrement) of a semi-circumference. There is an identical case 

 in the spherical pendulum. Regarding plane as the final limit of narrowing spheri- 

 cal vibrations, it is evident that the azimuthal velocity of passage by the vertical 

 becomes very great and has its limit infinity when they pass tJirougJi the vertical. 



This expression ^(equal to j3 f| nearly) is a very different .thing from the 

 "mean precession" of the gyroscope, o#Jf' g iven in mv analysis. The latter is 



