30.2 Royal Society : — 



of 5 to the positive axis of jy), or in the same direction as that of the 

 rotation w ; and the terras A represent a circular motion, of period 



— , in the contrary direction. Now, to being very great, p and a are 



p 



very nearly equal to one another ; but p is rather less than o-, as the 



following approximate expressions derived from their exact values 



expressed above, show : — 



,1 \^ 1 X^ .1 V ,1 \^ 



8w-n Sw^ Swii 8(1)^ 



Hence the form of solution simply expresses that circular vibrations 

 of the pendulum in the contrary directions have slightly different 



periods, the shorter, — , when the motion of the pendulum follows- 

 <r 



27r ... 1 



that of the arm supporting it, and the longer, — , when it is m the 



P 



contrary direction. The equivalent statement, that if the pendulum 



be simjpJy drmon aside from its position of equilibrium, and let go 



without initial velocity, the vertical plane of its motion will rotate 



slowly at the angular rate — (a—p), is expressed most shortly by 



taking A=B, and reducing the preceding solution to the form 



4= 2A cos tat cos n't, 



r}=2A sin zyt cos n't, 

 where 



1 I \* 



n'=-{ff + p), or, approximately, n'=n + — ~:;~, 



2 o w'n 



and 



1 . 1 X'' 

 'nr=:— (ff— p), or, approximately, -37^5 — =. 



2 8 (0 



It is a curious part of the conclusion thus expressed, that the 



faster the bearing arm is carried round, the slower does the plane of 



a simple vibration of the pendulum follow it. When the bearing 



arm is carried round infinitely fast, the plane of a vibration of the 



pendulum will remain steady, and the period will be m ; in other 



words, the motion of the pendulum will be the same as that of a 



2 

 simple pendulum whose length is — ^^ , or a harmonic mean be- 



-+- 

 e m 



tween the effective lengths in the two principal planes of the actual 

 pendulum. 



It is easy to prove from this, that if a long straight rod, or a 

 stretched cord possessing some rigidity, unequally elastic or of 

 unequal dimensions, in different transverse directions, be made to 

 rotate very rapidly round its axis, and if vibrations be maintained in 

 a line at right angles to it through any point, there will result, 

 running along the rod or cord, waves of sensibly rectilineal trans- 

 verse vibrations, in a plane which in the forward progress of the 



