AS ]: x ii i I;ITIN<; THE ROTATION OF THE KAKTII. 27 



which denotes that the line of apsides moves through ;i quadrant while the pendu- 

 lum is passing, through ISO' 1 azimuth, from a highest to a lowest point; in other 

 words, that the highest and lowest points are diametrically opposite, and the apsides 

 are ujijHUTiilly stationary. This theorem is true (as shown by Prof. Peirce), what- 

 ever be the inclination of the great circle, though it cannot be generally made evi- 

 dent by the above formula-. 



If in (r) we make transformations and substitutions already described, develop 

 the factor (p )"', integrate between the limits =a, =/, and double the result, 

 \v< shall have for the time of one vibration, or one semi-orbital revolution, 



and if a=0, that is, if the motion is purely oscillatory, then p=2/, and this becomes 



and more generally for any value of a and ft not exceeding the versed sine of thirty 

 or forty degrees: 



When a and t i are both small, as in most pendulum experiments, the terms after 

 the first in the brackets may be omitted, and we have, the ordinary expression for 



the time n r 



N tf' 

 In the expression (gr), omitting all the terms in brackets after the second, it is 



evident that that term will measure the apsidal motion for the timen I-; and hence 



**9 

 that the total integral of equation (a) taken through that time will be, 



and if we denote by <p' the angle of azimuth of the apsidal line measured from its 

 initial direction, we shall have, at any time t, substituting for a and ft, I (1 cos 0J 

 and I (1 cos 2 ),* 



(&) $':=[n sin 2, + *" P (1 cos 0J (1 cos 0j) 



L 4\l 



or, since 0, is always small, 



V) <p'=[n sin aLl /ni ~oi"' l/2sin 



The angle 2 being given, 0, is determined for the ordinary pendulum experiment 

 by the consideration that the constant C, of which the value is found p. 24, is the 

 moment of the quantity of motion. As c is very minute, the actual velocity at the 



