26 



ie) 



THE PENDULUM AND GYROSCOrE 



-cons t. 



In onlcv that tlic arcs in tlie ahove expression should continually increase with 

 tlie time, the positive or negative sign must be applied to the radical ^Z U accord- 

 ing as u is increasing or diniinisliing: taken from »=a to ?(=/3 (or the converse), 

 tlie arcs all become =7t, and the non-circular functions vanish. 



Hence the azimuthal angle passed over by the pendulum in its motion from a 

 lowest to a highest point of its orbit (or the converse) is expressed by 

 1 r, 1 _ /I , 1\ , 1 ,— /\ ,1,3 \a+/3 



1 /I ,1,3 ,15 \/3(a+/3)- 1 ,j \ 



-I /-T./I1 1_L S , 15 105 w5(«+/3)' M(^+^)U&c"l 



Tlie sum of the terms after unity included in the brackets is the ratio by which 

 the azimuth angle exceeds a quadrant; or, if the integral is taken through an entire 

 revolution (relatively to tlie a2>sides), it, multiplied by 27t, is angle -of advance of the 

 apsides per revolution. 



For motion nearly oscillatory, of whatever amplitude {i. e., a being small and /3 

 arbitrarily large), or for spherical motions of considerable amplitude (a and (3 taken 

 witliin limits not exceeding say one-third of 7, corresponding to a swing of over 

 UU°), p, always greater than 21, differs but slightly from that magnitude. Giving 

 p tliat value, and tiiJving tlie angle ^ for a complete vibration, or a semi-apsidal 

 revolution, we have the formula. 



If a and ^ are both small and nearly equal, and d the angle of which they are 



the versed sine, then 1^=1 sin^ 6, and the apsidal motion corresponding to the 



second term of the above (the following terms neglected) becomes f 7t sin^0; agree- 

 mg with the expression (a), on p. 24, when developed for the same case. 



If the pendulum moves nearly horizontally in a great circle, that is, if a-\-^=2l 

 andaf3=P (nearly), then p, C, and r are each infinitely great, and (/) becomes 



t,.=i.[i+i+|+_i_+i_+&,]^, 



