30 



THE PENDULUM AND GYROSCOPE 



reduce to the constant r' {v being tlic impressed angular velocity of pendulum 

 movement), and the second and third of equations (3) will yield the cquatioii- 

 _'^'C,, cos ?.— 2 sin X)=—v- (y cos ?.—z sin X)— i/ sin Ti 



the intcEjral of which is 



y cos A-2 sin ;^=_^'1J^+^ cos vt-\-B sin vt 



The above is independent of n; y cos 1—z sin X is tlie value of the new co-ordi- 

 nate of y' wlien the axis of // is changed to parallelism to that of the earth. Hence, 

 as tlius transformed, this co-ordinate is unaffected by the earth's rotation, the plane 

 of pcnduhnn motion must turn (if it turns at all) about such an axis. 



The assumption for ^=0, of //=(), z=l, '^ji—lv, -,/=0, gives 



at dt 



B—l cos ;i 

 (10) hence y cos X—z sin A= — ^? sin A (1— cos i-t)-\-l sin {vi—7C) 



The second term of the second member of the above gives precisely the value 

 which the first member would have, were the plane of pendulum motion stationary 

 or were it turning Avith any angular velocity about an axis parallel to the earth's. 

 Tlie first term is, owing to the assumed high value of r, very minute, and is periodic, 



the period being equal to , the time of the pendulum's revolution in its circular 



V 



orbit. Owing to its minuteness and periodicity, this term may be neglected, and 

 equation (10) becomes 



y cos ?, — 2 sin "k^l sin [vt — ?.) 



to satisfy which, and at the same time the three differential equations (3) (omitting 

 g, as we have found reason to do in (10), and also omitting terms containing iu, in 

 the developments), requires the following values for the co-ordinates : — 

 x=l sin nt cos (^vt — A) 



y—l [cos X sin («<_;\,)-[-sin 7. cos nt cos {vt—X)^ 

 z=:l [—sin ?i sin (r/_a)-}-cos A cos 7d cos (v<— X)] 



Changing tlie axes of y and z by turning them through the angle X, we should 

 have for new co-ordinates, 



a;'=/ sin nt cos (vt — X) 



y'=I sin (r/— X) 



z'=I cos nt cos (vt — X) 



If we now change the plane of yz by moving it through the variable angle 7if 

 about the axis of y\ we get, 



x"=0 



y"=I sin (r/— X) 



^'~l cos {vt — X) 



y'^-\-z"'^P 



