188 V. OSCILLATIONS AND CYCLIC MOTIONS. 



Proceeding in the same manner for 77, we have with the same 

 degree of approximation 



d$ ml* , c 



W = "~^ n "V 

 181) 



If W is the potential energy (there being no apparent potential 

 energy due to the cyclic motion, since the part C is here constant), 

 the equations of motion are accordingly, 



182) 



Thus the gyroscopic terms in c have the property proved in 172). 

 If there is no potential energy, the gyroscopic forces cause the 

 motion to be of such a nature that 



rr + tf'i/ = o, i/i" 2 +v 12 - ^v^+^' 



that is the acceleration is perpendicular to the velocity, and pro- 

 portional to it. Under these circumstances the motion is uniform 

 circular motion. In fact the equations are satisfied by 



I = Acospt. v c 



183) W = 0, p = - 



t] = A srnpt, ml* 



Thus the circle, whatever its size, is described in the same time 

 ~> which is inversely proportional to the momentum of the cyclic 



motion. We may describe the effect of the gyroscopic forces in 

 general for a system with two degrees of freedom by saying that 

 they tend to cause a point to veer out from its path always toward 

 the same side. This effect is characteristic, and cannot be imitated 

 by any arrangement of potential energy whatever. By the aid of 

 this principle all the motions of tops and gyrostats may be explained. 



51. Cyclic Systems. A system in which the kinetic energy 

 is represented with sufficient approximation by a homogeneous 

 quadratic function of its cyclic velocities is called a Cyclic System. 

 Of course the rigid expression of the kinetic energy contains the 

 velocities of every coordinate of the system, cyclic or not, for no 

 mass can be moved without adding a certain amount of kinetic 

 energy. Still if certain of the coordinates change so slowly that 

 their velocities may be neglected in comparison with the velocities 

 of the cyclic coordinates, the approximate condition will be fulfilled. 

 These coordinates define the position of the cyclic systems, and may 



