105, 106] 



FOUCAULTS GYROSCOPE. 



325 



and at right angles to it, like the axes , y, g of Fig. 112. We then 

 have for the motion of the axes 



254) ^ = -&sin#, q = d ~> r = 



while if to be the velocity of rotation of the top about the ^-axis, 

 we have for the moment of momentum, the axes being principal 

 axes, though not fixed in the top, 



255) 



H x = A& sin #, H y = A 



H z = Co. 



~ dt 



Inserting now in the equations 29), 84, the constraint producing 

 a couple Z, 



d& 



~dt 



256) 



+ 



n do 

 Q -77 



at 



. 



--J-- -f 

 dt 



~ 



-^- = 0. 

 dt 



From the last of these equations, o is constant, while from the 

 second, neglecting i 2 , we have 



The first equation 256) determines the constraint L. Equation 257) 

 is the equation for the motion of a plane pendulum, 22, so that 

 the gyroscope will perform oscillations about a line parallel to the 

 earth's axis, or will be in equi- 

 librium when -9 1 = 0, thus afibrd- 

 ing a means of determining the 

 latitude. The time of a small 



oscillation will be, 2 



which, on account of the small- 

 ness of &, will be very great 

 unless o be made very great. 

 The experiment was performed 

 with success by Foucault. 



In the second case let us 

 suppose the gyroscope con- 

 strained to move in a horizontal 

 plane. Let us take for IT- axis 

 the vertical, corresponding to 



the -axis of Fig. 112, for the Z-axis the axis of figure of the 

 top, making the variable angle (p with the north, towards the 

 east, and for the X-axis a perpendicular to these (Fig. 113). The 

 rotation of the earth gives the components & sin #, & cos # in the 

 direction of the , g axes respectively (# being the co- latitude and 



Fig. 113. 



