350 



Choosing tMs vertical for axis of (%) and the axis of (x) in vertical 

 plane through the axis of the gyroscope, the components of the earth's 

 attraction on any element dtn are easily seen to be 



Neglecting terms with coefficients — . 



where R = the radius of the earth. 



1 



R^ 



.-. moment round the axis of (y) = 2 ( X - xZ) dm) 



= - 2 %xdm. 

 R 



To determine this, let z'x' be the co-ordinates with respect to the axis of 

 the gyroscope, and a line at right angles to it in the same vertical 

 plane, the axis of {y) being left unaltered ; then 



r (io% V - x' sin V, 

 \x = %' ^in V ■\- x' cos V, 



when V = inclination of the gyroscope to the vertical ; 



if = - ^ sin 1^ cos V ^dm {z^^ - x'^), 

 since '^dm (%'x') = 0, 



or — sin v cos v{C - A)., 



R 



this moment (M"), acting downwards in the vertical plane passing 

 through the axis of the gyroscope, will be the sole effect of the earth's 

 attraction. It will produce terms in the equations with a coefficient 



L 

 R 



These terms will be, of course, inappreciable when compared with the 

 terms whose coefficient is (ww) ; but they will be far greater than the 

 terms which have (w^) as a factor. We cannot, therefore, in these 

 equations make {m) equal cypher, and assume that the result will re- 

 present what happens when the gyroscope is started without any motion 

 round its axis. 



All such conclusions would be based on the im.aginary hypothesis of 

 the equality of the earth's pttraction at different points of the gyro- 

 scope. 



That the inequality of attraction would materially affect the result 

 when the velocity of the spin is of the same order as (w) may be shown 

 as follows : — Supposing the gyroscope placed in its frame without spin. 



