AS EXHIBITING THE ROTATION OF THE EARTH. 23 



the mean azimuthal motion of the body itself, the former that of the nodes of its 

 orbit. The hitter is strictly true only for very great values of fS ; the former, 

 rightly interpreted, is always true, though it has no special applicability except for 

 (as in the gyroscope pendulum) small values of /3. The harmony of the two 

 expressions is easily shown. 



Practically the gyroscope is made up of not only a rotating disk, but a non- 

 rotating frame. In estimating the "deflecting force," therefore, in the expression 

 (a) C should apply to the disk alone, and 31 and y to the entire mass of disk frame 

 aild stem. 



The solutions that have been given of equations (4)., for the freely suspended 

 pendidum are restricted to very small motions ; the following is general. 



Transfer equations (4) to polar co-ordinates by substituting for x, y, z the values' 



.7=:? sin ^ sin 

 y^l cos 4) sin 

 z =^l cos B 



in which ^ denotes the azimuth of the pendulum measured from the north, and Q 

 its deviation from the vertical, and we get 



( -T^='i -sill A-4- „ . ^ „ .-- o^ ( cos d) sur B dB 



at ' I' sm- d sur- Q J 



in which C is a constant depending on arbitrary initial values of ^, the final term 



dt 



corresponding to the last term of (4). 



At the equator we have ?„^0, and the azimuthal velocity expressed by the third 



term of (7) becomes ^,-- fcos A sin" Q dd, which being periodic, produces but 



snr^v/ 



very minute change in the plane of vibration. If the pendulum is propelled, 



from a state of rest in the vertical, in the direction measured by the angle <^ 



from the meridian, this angle will be but very slightly aftccted by the minute 



values of the above expression during the outioard excursion, and the increment 



wliich '? receives Avill be almost exactly neutralized {quite so if cos ^ were abso- 

 dt 



lutely invariable) during tlie return, and the angular velocity due to the term will 

 again become zero; which cannot happen unless the pendulum again pass through 

 the vertical on its return, in which case ^ will be as little varied during the return ; 

 (otliericise ^ will, during the return, pass through all possible values from to ^tt, 

 and integration is impracticable). Hence we may assume <p as constant, and, as in 

 any other latitude, tlie term in question is, multiplied by cos ?., the same as at the 

 equator, Ave may generally integrate that term for plane vibrations, considering 

 ^ constant, and putting C=0. Equation 7 thus becomes, 



,ox d^ ■ „ „0 — sin cos 



(8) -i=H sm X — 71 cos X . „ 



^ ^ dt sin^e 



' The following analysis, as far as equation (9), is modified from Galbraith and Houghton. 

 Proc. R. I Acad. 



