AS KXlIir.ITING THE ROTATION OF THE EAUT1I. 23 



the mean a/imuthal motion of the body itself, the former that of the nodes of its 

 orliit. Tin- latter is strictly true only for very great values of (3 ; the former, 

 rightly interpreted, is ;il\vass true, though it has no special applicability except for 

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

 expressions is ea-il\ >hmvn. 



1'ractically the gyroscope is made up of not only a rotating disk, but a non- 

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

 (a) (' should apply to the di>k alone, and J/ and y to the entire mass of disk frame 

 arid stem. 



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

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



Transfer equations (4) to polar co-ordinates by substituting for x, y t z the values 1 



x=l sin < sin 

 y=l cos <p sin 

 z =1 cos 



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

 it> deviation from the vertical, and we get 



(7) =n sin ?. + ___-.rcos $ sin 1 d 



di ' / Mir sm 1 J 



in which C is a constant depending on arbitrary initial values of S3, the final term 



at 



corresponding to the last term of (4). 



At the equator we have /l=0, and the azimuthal velocity expressed by the third 



term of (7) becomes ' f cos $ sin 2 dO, which being periodic, produces but 



sin QJ 



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

 i'mm a state of rest in the vertical, in the direction measured by the angle q> 

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

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



which * receives will be almost exactly neutralized (quite so if cos d were abso- 

 dt 



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

 attain 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 ; 

 (otherwise $ will, during the return, pass through all possible values from to :t, 

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

 any other latitude, the term in question is, multiplied by cos A, the same as at the 

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

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



(8) **=n sin a-n cos 



dt 



1 The following analysis, as far as equation (9), is modified from Oalbraith and Ilonghton. 

 Proc. R. I. Acad. 



