128 THEORY OF NEWTONIAN FORCES. [PT I. CH. III. 



rigidly fastened to the box. If the box be at rest, there is nothing 

 on the outside to lead one to suspect the presence of the fly-wheel, 

 but if the box be moved about the reactions developed will be very 

 different if the concealed wheel is in rotation or not. In the 

 former case the least experimenting will render us sure of the 

 existence of a concealed motion. To convince oneself of the truth 

 of this statement it is necessary only to take a toy gyroscope in 

 one's hand and turn it about. 



67. Cyclic Motions. In certain cases some of the co- 

 ordinates do not appear in the expression for the kinetic energy, 

 although their velocities may. Such are termed by Helmholtz* 

 cyclic coordinates, and we shall distinguish them by a bar. The 

 example just given of the gyrostat is such a case, for the angular 

 coordinate fixing the rotation of the fly-wheel does not appear, but 

 only its derivative, the angular velocity. Further examples are 

 furnished by the case of a heavy belt running over pulleys, or by 

 the case of a fluid circulating in an endless tube. The coordinate 

 expressing how far a point on the belt or in the fluid has travelled 

 does not enter, but its velocity does.. The condition for a cyclic 



7\T 

 coordinate being = we have for the forces maintaining cyclic 



motions [ 61 (7)], 



If 



dt \dq 8 ' 

 If the forces of the cyclic motions vanish we have 



lf 

 dt \dq s ' 



or, integrating, 



In this case we may with advantage apply Routh's transformation 

 in the case of the cyclic velocities. The equations for the elimina- 

 tion are now 



Qn5i' + Qi2<? 2 ' ... + Qir<lr = Ci~Qi r+iq'r+i - Qin^n', 



* Helmholtz, Studien zur Statik monocyklischer Systeme, Borchardt's Journ. 

 fur Math., Bd. 97. Wiss. Abh. in. p. 128. 



