100] 



STEADY MOTION OF ROLLING HOOP. 



309 



to it. It is to be observed, either by considering this example, or 

 from the results of 90, that in order to prevent the hoop from 

 falling, it must be steered, or given a pivoting movement, towards 

 the side to which it tends to fall, and this is the practical manner 

 of steering a bicycle. It is to be remarked that in steering the 

 bicycle by the rider, the centrifugal force plays a greater part than 

 the gyroscopic action of the wheel. For a treatment of this subject, 

 the reader is referred to Appell, Traite de Mecanique Bationelle, and 

 Les Mouvements de Eoulement en Dynamique, and to papers by Bourlet, 

 Carvallo, and Boussinesq. 



We will now consider the general treatment of the motion of a 

 body bounded by a surface of revolution, and dynamically symme- 

 trical about the axis of revolution, rolling 

 without sliding on a rough horizontal plane. 

 We shall follow the method and notation of 

 Appell. Let us take as axes, with origin at 

 the center of mass of the body, as in 90, a 

 set of moving axes turning about themselves 

 with angular velocities p Q , q , r , of which the 

 ^-axis is the axis of revolution, the Y-axis 

 the horizontal axis in the equator of the body 

 (the line of nodes of 90) and the X-axis 

 directed toward the ground in the vertical 

 plane containing the ^-axis (Fig. 110). We 

 have accordingly to put in Euler's geometrical 



equations, 65), (p = - so that 



Fig. no. 



These are connected with the rotation of the body by the relation 



(cp not being constant for the body). 



For the motion of the center of mass, we have the components 

 of the weight of the body 



together with the unknown components of the reaction, which we 

 will call R x , Eyj E z . The resultant is to be equated to the product 

 of the mass by the acceleration of the center of mass, using the 

 method of 77 for moving axes. If v x , v y , v z are the components of 

 the velocity of the center of mass along the instantaneous positions 

 of the moving axes, we have accordingly, substituting v x , v yj v z for 

 x, y, z, in 128) 77, 



