ON THE RESISTANCE OF ROAD VEHICLES TO TRACTION. 317 



(b) Simple rotation of a cylinder upon an horizontal surface, when 

 the surface formed on the plane is an envelopment of the cylinder. 



A cylindrical body alone can roll upon a plane in a straight line. If 

 the body be not cylindrical, the rolling motion is accompanied by a sliding 

 one, or, what amounts to the same thing, a pivoting motion. 



This action is particularly noticed when a wheel with a cylindrical 

 tyre leaves one straight line direction for another, and such a wheel, when 

 rounding a curve, is subjected to simple rotation and also a pivoting 

 action about its vertical axis, and consequently tends to twist the wheel 

 from its true position to take up a position relative to the curve. 



Morin experimented in the following manner : Upon two parallel 

 beams, with horizontal faces, were placed two loaded cylinders. In order 

 to give motion to these cylinders additional weights were added, acting 

 tangentially to the cylinders and parallel to the beams. 



For different cylinders and surfaces the force required to impart 

 motion was fount! by pxpe.riment. . 



The force thus found, which imparted a uniform rolling movement to 

 the cylinders, must be that which just overcomes the resistance to rolling, 

 for if the force were too great, the motion would become uniformly 

 accelerated, and if the force were too little, the cylinders would come to 

 rest. 



Consequently, from the above it may be inferred that the moment of 

 the additional weights about the axis of rotation is equal to the moment 

 of the frictional resistance about the same axis. 



Calling R the resistance due to friction, which is supposed to act 

 tangentially to the wheel, of which ?• is the radius, ^9 the pressure trans- 

 mitted by the wheel to its supports, and A a constant ; then we have 



R = Af- 



r 



which sums up the law for rolling friction. 



From the experiments, which were made by Morin to ascertaiti the 

 truth of Coulomb's Law, he drew the following conclusiohs : — 



(i.) On fibrous materials, such as wood ; on spongy textures, ds leather ; 

 on gi-anular bodies, as plaster : ' That the resistance to rolling varies 

 inversely as the diameter of the wheel, measuring this fdrce (R) at the! 

 circumference of the wheel.' 



(ii.) On compressible surfaces : ' That the resistance to rolling ihcreases 

 as the width of the tyre diminishes.' 



(iii.) On an elastic substance, such as indiarubber : 'That the depth of 

 the depression is practically proportional to the pressure, provided that 

 the elasticity remain constant ; the depth of the depressions increases as 

 the width of the tyre diminishes. The elastic reaction, or the force with 

 which the substance tends to return to its original form (after the pressure 

 has been removed) is not always very rapid even in the case of india- 

 rubber, so it is much less so on a stone road ; in consequence, it does not 

 restore to the wheel the whole force expended on it. Even on railways, 

 where the rails return very quickly to their normal shape, the speed of 

 the train is too great for the vehicles to recover the work that has been 

 transmitted to the rails.' 



(iv.) On a homogeneous body, such as wood : 'That the pressure alters 

 the elasticity, and that the resistance to rolling increases more rapidly as 



