22 TlIK PENDULUM AND GYROSCOPE 



Develop cos 0=-(l-siir 0)' and we get 



d^_ Cn A I 2 lz:^_i siir 0-i sin' 0-&c.) 



For ordinnvy ranges of pendulum vibration the terms involving positive powers 

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

 omitted, and wc have 



di~~~~ "lA^ A sin=0 



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



Cn 

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



term Avhich represents the angular velocity in the oih'd. Indeed we have, in the 

 second term, the motions of tlie 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 is indefinitely small, and c must hence 



dt Qi^ 



be unity. Hence we see that at the very outset the initial value -^^ "^^ist be attri- 

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

 dt 



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



through tlie vertical at every return. The horizontal pro- 

 jection of the curve would be a series of loops radiating 

 from a common centre. For each complete vibration 



the intenrral of '^--dt would represent the entire angular 

 2A 



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

 gram) from .4. to A, &c., and the integral of the remain- 

 ing terms should be 'in. 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 tlie point of suspension. 



Though the numerator of the fraction —-^^-\ becomes zero for the case iust con- 



A sin^ ■' 



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

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

 time of passage nuist 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 iJiroiujJi the vertical. 



This expression ^jl^^l"'^! ^^ /^J^ nearly) is a very different thing from the 



■mean 



precession" of the gyroscope, -^^W, given in my analysis. The latter is 



