jg THE PENDULUM AND GYROSCOPE 



Since ^"-+v/^+2==r, Avc have x <?x+Z/ <b+^ <^'=0- Hence multiplying equa- 

 tions (3), respectively, by dx, dij, and dz, and adding, wcliave 



dx cPx-\-dy dhj-^dz dh _ 

 ' cW -^ 



dx^-ird>f+dz^ ^ , 

 (3). -^-dl^'^-—^'+' 



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 heiglit of f dl. 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, tliat tlie forces wliich maintain this relative fixedness are equivalent to a force 

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

 to tlie patli. 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.^ 



IMuUiplying the first of equations (3) by v/, and the second by x, and adding, we 

 get:— 



d^x d"i/ c ■ ■,/ di/ , dx\ , o ^ ^^2 



Integrating : — 



(4) y^—x'^jL=n sin 7. Oi>'+?/")+C+2/i cos T.jyaz 



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 

 ^alue) increased by what is due to the constant angular motion n sin X, and by the 



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



» Tliis conclusion is not invaliiiated by flie introdiu'tion of tlie disturbing forces of the ordor /)' 

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

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

 will thence be 



2 rrl sin 2ji 

 1V I cos^ X 



The third of these is an incremrnt to gravity, and the first tends to prolong vibrations 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 inappreciabbj minute, the 

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



1.800.000.000' ''•'''"■^''^'■"S "'c time of vibration by about ^j- ]^^ . The first has the con- 



o.bOU.OOO.OOO 



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

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

 ina.Minura, ' ' 



