24 On the Nalure and Laws of Friction. 



have p = ~7^I — r = the power that would overcome 



1 5ST4 Ri x4Jx 13 



the resistance at the ciicumference. 



And, (Z + p = -n + r—x — 2 = t"e force or power 



'""' 1-5874 lO X i^ x„3 



that would keep the wheel in motion. 



The value of the constant n may be derived from experiments 

 on the friction of the axis ; and when the value of??, is previously 

 determined, the value of q might be obtained from experiments 

 on wheel-carriages. But as q will differ according to the nature 

 of the road, its precise value is not of much importance ; and 

 particularly where a rigid analysis of the subject has not been at- 

 tempted. All I have endeavoured to do, is to show an useful ap- 

 proximation to the laws of resistance according to the radius, 

 the breadth, and the velocity of the wheel, the radius of the axis, 

 and the weight of the load. 



-- . 1 . I . n X r X W /rj /-, % 



Comparmg the resistance at the axis, — , (rrop. i^.j, 



with the resistance at the circumference -rr-^ — , (Prop. G.), 



it appears that these resistances will be equal when = 



■^—^ — . Now, if we consider — to be equal '25, which will be 



nearly the true value of the friction in wheel-carriages, then tan. 

 Be="5r. In general the depth of the impression will be greater 

 than corresponds with this value of the tang, of Be; and when- 

 ever it is greater, the resistance at the circumference will exceed 

 the resistance at the axis. It would be desirable that some obser- 

 vations should be made on this subject, as then we could better 

 compare the effects to be gained by any variations in the con- 

 struction of wheels. 



It must be remembered, that in estimating the resistance at 

 the circumference, the surface of the road has been supposed to 

 be an uniform horizontal plane ; but granting the road to be ho- 

 rizontal, it will in practical cases be irregular, or covered with 

 small asperities. Now, the effect of these asperities will bear 

 some proportion to the number of them that the wheel is raised 

 over in a given tin e, which will depend on the breadth of the 

 wheel; conseciuently a broad wheel will increase instead of lessen 

 the friction, unless the road be perfectly smooth. 



From the preceding investigation the following conclusions are 

 drawn : but it may be necessary to remind the reader that they 

 apply to such roads only as are nearly horizontal. 



I. The resistance at the axis is directly as the weight of the 

 load. That is, if the load be doubled, the resistance at the axis 

 will be doubled j and so on. 



