THK C.VROSCOPE. 63 



then fall, gain velocity and reverse its curvature, finally mov 

 ing horizontally at the bottom of the loop. The other half of 

 the loop will be similar and the curve as before will be a pro- 

 late cycloid, but with the summits at a greater elevation. If 

 the initial velocity be made still greater and equal to twice the 

 critical value, it can be shown mathematically that the com- 

 mon cycloidal motion will be again resumed with the bottom 

 of the loops at the level of starting. The motion being 

 cycloidal once more, we know that the initial velocity must 

 have been the same as the horizontal velocity generated at the 

 bottom of the loop in the case where the axis starts from rest. 

 Hence the critical value is one-half the horizontal velocity 

 which is attained at the bottom of a loop when the apparatus 

 starts from rest. If an initial velocity still greater than twice 

 the critical value be given to the axis, the curve, as can be 

 shown by analysis, will be a curtate cycloid with the small 

 loops uppermost, the summits being at a higher elevation than 

 ever. If, to pass to the other extreme, the axis is started with 

 an initial velocity directed backward, we will find described 

 another curtate cycloid with the small loops upward, the sum- 

 mits being at the level of starting. 



The action of the theoretical gyroscope having been inves- 

 tigated, it now behooves us to ascertain wherein the practical 

 instrument differs in its action from the theoretical. As has 

 beerf intimated before, the principal modifying factor is fric- 

 tion, tending as it does to slow down the rotation of the wheel 

 and to oppose the orbital motion. 



The results of the modif'ying influence of friction may be 

 reduced to two classes : ( a ) Those due to the decrease of velo- 

 city of rotation of the wheel, and (b) Those due to forces 

 opposing the orbital motion. 



(a) Perhaps the most important effect of friction is found 

 in its opposing the rotation of the wheel. It is necessary to 

 examine in detail the action of the apparatus as the rotational 

 velocity becomes less and less. As we have already seen, 

 since the deviating force is alwavs at right angles to the direc- 



