98, 99] MOTION OF BILLIARD BALL. 305 



account of friction. The friction of sliding is supposed to be a force 

 of magnitude F= pR where R is the reaction between the ball and 

 the table, and ^ the coefficient of friction. F has the direction 

 opposite to that of the motion of the point of contact of the ball 

 and table. 



If the axes of X, Y are taken horizontal , Z vertical, we have 

 for the motion of the center of mass the equations 



~ = X = Fcos (Fx) = [iR cos (Fx), 



178) 



and since is constant, R = Mg. 



Euler's dynamical equations are, since A = B=C, 



179) 



a being the radius of the sphere. To determine the direction of JE 

 we have 



IT V 



180) m f-^=, 



where v x , v y are the velocities of the lowest point of the sphere, 



Differentiating these equations, and making use of 179), 



*v x _d*x d ^_^ J _ ai ^ 

 'dt~W a dt~ M" A 2 



182> S-3 + .Jf-i+5'- 



Dividing one of these by the other, and using 180), 



dv x dv y 



20 



from which 



WEBSTER, Dynamics. 



