AS EXHIBITING THE ROTATION OF THE EARTH. 21 



may be disregarded. To prove this, I refer to iny expression (/i) and the context, 

 in ray analysis of the "Gyroscope."' 



, , Cn 



(") -yM ^^ 



for the deflecting force, as. I call it (a force due to the rotation of the disk with 

 angular velocity n, and acting, at the centre of gravity, norraully to the plane of 

 angular motion of the disk-axis, or of the arm of the gyroscope pendulura), in 

 which M is the mass, C its moment of inertia about the disk-axis, y the distance 

 from its centre of gravity to point of suspension, and 'i\ the angular velocity. 



Disregarding the vertical motions represented by on account of the smallncss 

 of the arcs, - - and y -,y (I substitute I for the y mentioned above) would repre- 

 sent very nearly the components of angular velocity of the centre of gravity. 

 Substituting these for v^ we shall get the components of the "deflecting force," 

 and the equations of motion will be, 



d'x Nx _ C)i dy 

 ^t^~f'~~fM di 



d^y Ny Cn dx 



~M^~f^^¥Mdt 



d?z . Nz , 



dt ' I 



These equations are identical in all but the value of the coefficients with (3), 

 when transformed to (8)2, under the same license. Of course the motions of the 

 gyroscope pendulum would have the same solutions, the mean azimuthal motion of 



the nodes of its orbit being expressed by half the coefficient of - ^, — ,r„ ,>, or, since 



dt 2rM 



the moment of inertia A, of the gyroscope, with reference to a principal axis 



through the point of support, is (Z being supposed to be very large compared to the 



dimensions of the disk) very nearly PIT, the mean azimuthal motion is more simply 



expressed by - - . 



This may be more generally proved as follows : Tlie first of the general differ- 

 ential equations (equations 4 of my analysis) of gyroscopic motion is 



sm- d -^-=-— (cos — c) 

 dt A^ 



in which 0, counted from the inferior vertical, is the variable inclination, and 4- the 

 azimuth angle of the disk-axis or pendulura arm, and c a constant depending on 



initial values of Q and - . 



dt 



" Sec American Journal of Science, 1857, and Barnard's American Journal of Education, 1857. 



