26 THE PENDULUM AND GYROSCOPE 



+1 



|-[-&c.,-(-cons't. 



In order that the arcs in the above expression should continually increase with 

 the time, the positive or negative sign must be applied to the radical |/ U accord- 

 ing as u is increasing or diminishing: taken from u=a to u={3 (or the converse), 

 the arcs all become =n, 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 



The 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 tJie apsides'), 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/? taken 

 within limits not exceeding say one-third of Z, corresponding to a swing of over 

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

 p that value, and taking the angle q> for a complete vibration, or a semi-apsidal 

 revolution, we have the formula, 



If a and ft are both small and nearly equal, and 6 the angle of which they are 

 the versed sine, then V^\ sin 2 0, and the apsidal motion corresponding to the 



I 



second term of the above (the following terms neglected) becomes f n sin 2 6; agree- 

 ing with the expression (a) 2 on p. 24, when developed for the same case. 



If the pendulum moves nearly horizontally in a great circle, that is, if a+/3=2Z 

 and ap=P (nearly), then p, O, and c are each infinitely great, and (/) becomes 



