ON THE RESISTANCE OF ROAD VEHICLES TO TRACTION. 327 



Here he deals with the loss of energy in mounting obstacles, and 

 shows that this loss is given, when h is small, by the following : — 



WV- 2/i 



Loss of energv=: X . . . . (4) 



°^ cj r ^ ' 



This is lost while the vehicle is moving a distance:=2^, say, and hence 

 he shows that the average effort is given by 



Average eflFort= — '' . . • (5) 



g r x/ 2r — h 



He concludes from these formulae that the draught on a road whose 

 surface presents a number of unyielding and projecting obstacles varies 

 directly as the square of the velocity, and inversely as the square root 

 of the cube of the size of the wheel. He goes on to say that the draught 

 upon irregular surfaces must be much more than that which has been 

 ascertained, in all cases where the velocity of motion is considerable, 

 because there is no substance perfectly inelastic, and therefore, after 

 striking the obstacle, the wheel will rebound backwards from the obstacle 

 with a velocity proportionate to the relatioii between the forces of impact 

 and restitution referable to the elasticities of the wheel and the material 

 of the obstruction ; moreover, the obstacle when struck will, in many 

 cases, slide to some extent, and thereby weaken the momentum. 



He then deals with the tractive efl'ort required for a yielding surface, 

 and states that to a great extent resistance to rolling is due to a continual 

 displacement of a portion of the road material owing to its inelasticity, 

 which, while it allows that material to exert a pressure against the fore- 

 part of the wheel, will not permit it to rise behind, and thereby propel the 

 wheel by its reaction. Thus the draught upon a soft surface is much 

 greater than if it were upon a hard, unyielding surface 



He then deduces a formula for the draught upon soft surfaces, making 

 two assumptions : tirst, that surface of road is perfectly inelastic ; and 

 secondly, that the resistance to compression varies simply as the depth 

 compressed. 



Using the same notation as before, except that h is now the depth 

 to which the wheel sinks, the formula is 



P= ^^^'' . . . . (6) 



Area immersed * * " • V / 



or further p_3W Length immersed ._ 



8 Diameter of wheel 



and hence he shows that the draught varies inversely as the cube root 

 of the square of the size of the wheel, or little more than the inverse of 

 the height of the wheel. 



He then proceeds to prove that when R is the angle of friction of 

 rolling, A the angle of friction for the surfaces in contact at the axle, 

 d the diameter of the axle, and 2r the diameter of the wheel 



sin R -f tan A — 



P=W 1!_. . . . (8) 



1— sin R tan A 



2r 



