Om the Nature and Laws of Friction. 23 



be sensibly proportional to yxx*. Hence y x a; x Z- is as W, 



\v 

 or x: , as before. 



E. — When a wheel moves forward upon a horizontal plane with 

 any velocity v, the deptli of the impression will be expressed by 



yy.h-Av'-' 



The depth of the impression is as the force, and as the square 

 of the time the force acts; but the time is inversely as'the velo- 



w 

 city of the wheel : therefore x: — f ; which combined with the 



effect of the area cives x = ~. 



1/ X b X v" 



As the resistance which the part of the road immediatelv before 

 the wheel offers to the motion, arises from the pressure against 

 every point of the are Be; the sum of these pressures may be 

 considered as collected in one point of the arc, which point may 

 be called the centre of resistance. 



F. — The centre of resistance will he nearly at the horizontal 

 distance iy from the perpendicular C B. 



For the resistance at any point is proportional to the depth of 

 the impression at that point; and, assuming that the arc B c 

 does not sensibly differ from a straight line, it may be considered 

 proportional to the distance from c, and consequently to be col- 

 lected at the horizontal distance |y from BC. 



G. — At any instant of the motion of a wheel upon a horizontal 

 plane, the load W is to that part of the power which overcomes 

 the resistance at the circumference, as the radius is to the tan- 

 gent of the arc B e ; e being the centre of resistance. 



The lines Cb, I e and Ce, (fig. A.) are respectively parallel to 

 the directions of the weight, power, and resistance ; and therefore 

 constitute a triangle of which the sides are proportional to these 

 forces. But Cd : e : : radius : tan. Be ; therefore 



R ; tan. B e : : W : p ; where p is that part of the power 

 which is employed in destroying the resistance. 



In general, the value of x will be very small compared with the 

 other quantities : therefore to render the expressions less com- 

 plicated, the relation between x and y may be expressed by 

 y^ = 2l{x, and the tangent of the arc Be may be supposed to be 

 equal to {y. 



Substituting for x in the equation x = -''-, (Prop. E.),and 



making tan. Be= Jy in the equation - '^ ''"' ■', (Prop. F.), we 



• See Emerson's Fluxions, p. 2fi0, Ex. 17: 2d edition, 

 t I'hil. Miig. vol. liii. p. (i. Prop. (5.) 



B 4 have 



