18 THE PENDULUM AND GYROSCOPE 



Since ^ 2 -(-7/ 2 -(-z 2 =Z 2 , we have x dx-\-y dy+z dz=0. Hence multiplying equa- 

 tions (3), respectively, by dx, dy, and dz, and adding, we have 



dx d?x-\-dy 3?y-\-dz d 2 z 



~ d '' 



This expression is independent of n, and the velocity at any point of the path 

 depends, in the same way as does that of a pendulum vibrating over a motionless 

 earth, upon the height of fall. The plane of vibration of the chronometer pendu- 

 lum is maintained in a fixed relative position, thereby differing from a " freely sus- 

 pended" pendulum. It will be seen hereafter, in treating of the gyroscope pendu- 

 lum, that the forces which maintain this relative fixedness are equivalent to a force 

 varying directly as the angular velocity, applied at the centre of gravity, normally 

 to the path. Such a force will have no influence upon the velocity. Hence the 

 time of vibration of the chronometer pendulum is not affected by the earth's rota- 

 tion, nor by the azimuth angle of the plane of vibration. 1 



Multiplying the first of equations (3) by ?/, and the second by x, and adding, we 

 get: 



dz 



. . 



*- sm * n cos 



Integrating: 



(4) y- #-,- TZ sin A, (^M-?/) +<7+2/i cos ^ n 



\ ' i/ ,7j 7j V \ 9 f \ ./ 



yaz 



The above (4) expresses that the moment of the quantity of motion about the 

 axis of z is equal to a constant C (depending upon any arbitrarily given initial 

 value) increased by what is due to. the constant angular motion n sin A, and by the 



area 2 J ydz (in the case of ordinary plane vibration this is the projection on 



1 This conclusion is not invalidated by the introduction of the disturbing forces of the order ri 1 

 referred to in note to p. 17, for, since the arc of vibration of the chronometer pendulum is exceedingly 

 small, z may be considered as equal to I, the pendulum's length, and y as very minute. Those forces 

 will thence be 



T?X 



\ nH sin 2x 



n 2 I cos 2 x 



The third of these is an increment to gravity, and the first tends to prolong v.ibrations in the prime 

 vertical. The second is null in its effects, since, being always positive, it retards the vibration in 

 one direction as much as it accelerates it in the other. But they are all inappreciably minute, the 

 last being, for the seconds pendulum, an increment to the force of gravity at the equator of about 



-, decreasing the time of vibration by about The first has the con- 



1.800.000.000' 3.600.000.000' 



trary tendency to increase the time of vibrations if made in a prime vertical (or any other plane 

 than a meridian), but its effect is equally inappreciable even when (as in the prime vertical) it is a 

 maximum. 



