398 
PROFESSOR H. S. HELE SHAW OH THE THEORY OF 
Then must 
Q = P^ = Pcota 
otherwise the motion will be accelerated. 
Then, since the component of force applied on wheel A normal to the surface, in 
direction of arrow a, necessary to maintain adhesion, is balanced by the reaction at a, 
and also that at B by the reaction b', the effect of the two forces P and Q is the 
same as two forces at the axis of the sphere and two couples about the axis. The 
forces act respectively at p and q (fig 30). The couples P Xap and —Q Xbq. Their 
combined effect is 
M=J ) .ap — Q.bq=P(ap — cot a.bq) 
but 
ap=R cos a 
bq = H sin a 
Resultant couple about axis 
= P( Et cos a— R cot a sin a) 
= 0 . 
Therefore the only effect is that due to parallel forces P and Q acting perpendicular 
to the axis at the points p and q. The effect of these is equivalent to the couples 
(P.op+Q.og) acting in the vertical plane of the axis, and the force (P —Q) acting at 
the centre of the sphere. 
Figs. 30 and 31. 
In fig. 31 let 
y = angle of plane of rotation of the rollers with a vertical plane. 
Then supposing the rollers to be symmetrically distributed. 
(1.) Vertical force on point of contact of each 
P-Q 
2 
_P(l-cot«) 
2 
