22 On the Nature and Laws of Friction. 



C. — The eft'ect of the fiictien at the axis in retarding the mo- 

 tion of the wheel will be expressed by — : where W is the 



weight of the load, v the velocity, and n the ratio of the friction 

 to the pressure corresponding to some given velocity. 



Let a denote that part of the moving power which is employed 

 in overcoming the friction at the axis. The friction at the axis 

 acts with the leverage r in retarding the motion, and the part a 

 of the power acts with the leverage R : that is, in the case of 

 equilibrium, R xc= r x friction at the axis. And because the 

 motion of wheel-carriages is sensibly uniform, the friction is as 



W . n X W 



— *: or friction = , where n is a constant to be determined 



l)y experiment. Hence we have 



T-, n X r X W n X )• X W 



K X rt = or a = — „ . 



t' R X u 



The effect of the resistance at the axis is directly as the radius 

 of the axis, and inversely as the radius of the wheel : therefore, 

 when the resistance at the axis only is considered, it is an advan- 

 tage to make the radins»of the wheel as large, and the radius of the 

 axis as small as possible. Also, the greater the velocity the less 

 the friction. 



Of that Part of the Resistance of a Wheel- Carriage ivhich arises 

 from the Action of the Circumference upon the Road. 



It may be assumed that the external rim of the wheel is very 

 hard in respect to the road it moves upon, and that the load 

 will, in all cases, cause it to sink in the road. 



D. — When a wheel rests upon a horizontal plane, the depth of 



the impression will be nearlv expressed by —'. where W is the 



' - ' ■^ h X t/ ' 



weight of the wheel and load ; b the breadth of the wheel ; y the 

 ordinate of which the corresponding absciss is equal to the depth 

 of the impression ; and q a constant quantity to be determined 

 for each particular road by experiment. 



For the ultimate depth of impression is directly as the force, 

 and inversely as the area the force acts upon ; but the area is as 

 y xb, and the force is as W : therefore denoting the depth of 



the impression by x; x : , or a; = — ^, Otherwise, the 



quantity of matter displaced is proportional to the force which 

 displaces it. The area of the segment dcB, (fig. A.) multiplied 

 into the breadth of the wheel, will express the quantity of matter 

 displaced : and as the depth of the impression is always very 

 small compared with the radius of the wheel, the area dcB will 



• See Phil. Mag. vol. liii. p. 6. 



be 



