350 ON THE MOTION OF 



the friction to be a function of these velocities or of the press- 

 ure or of both conjointly. The effect of this would be to add 

 to the other accelerating forces three new ones applied to the 

 point (x,y,z,) of the form of 



, dx, — dx,, , dy, — rfu, , dz, — dz, 



v-^r 1 : *T' ^-V^' 



where <p is any given function of the pressure and velocity of 

 rasure dv, dv itself being equal to 



v [(.'to, — dx,y -+- (dy, — dy,y + (dz, — dz.y] . 



The pressure is then to be eliminated from the equations of 

 motion ; after which there will remain a number of equations 

 sufficient, in conjunction with the equations of the surfaces, to 

 determine the position of the body in terms of the time. 



If the friction, be the cause of it what it may. be exactly suffi- 

 cient to prevent all sliding, while it offers no impediment to the 

 body's revolution round the normal at the point of contact, the 

 motions will be of a nature much more resembling actual 

 oscillations and rotations on supporting surfaces, than in the 

 hypothesis of surfaces absolutely smooth, particularly when 

 the tangent plane at P remains throughout the motion nearly 

 horizontal. The effects of this kind of motion, of which the 

 pendulum with a cylindrical axis is the simplest possible spe- 

 cies, have not, that I know of, been examined by any author, 

 when the triple rotation of pitching, rocking and whirling are 

 all considered at once. Nevertheless, the problem of the small 

 i iscillations of the kind above described upon a plane or sphe- 

 rical surface is susceptible of complete integration and solu- 

 tion in the case both of free sliding and perfect rolling, what- 

 ever be the figure and constitution of the oscillating body, and 

 whatever be the velocity round one of the axes, provided that 

 it be compatible with small rotations round the other two. 

 I have given in the New York Mathematical Diary for July 

 1827 formulas which arc applicable to the case of all bodies. 



