294 On Friction. 



the phaenomena of friction, the kinds of motion are seldom, if 

 ever, distinguished ; and we are left to conclude that it is indif- 

 ferent whether the motion be uniform or accelerated. I hope I 

 have been successful in showing that there is a material differ- 

 ence ; and it appears to me that the distinction will be of much 

 use in many of the most important physico-mathematical re- 

 searches. 



Your correspondent further states, that the indentation cannot 

 be as the square of the time ; I know it is not correctly so ; but 

 from the following investigation it will appear that it is a very 

 near approximation to the truth. 



Let W = the weight or pressure producing the indentation ; 

 (^= the depth of indentation when the resistance of the 

 pressed surface is exactly equal to the weight VV3 



and x = any variable depth counted from the surface. 



Then, as the indentatior^ is always proportional to the pressure, 



d : a: : : W : — = the resistance at the depth x. 



Wx 

 w — j- 



Hence = 1 — — =the accelerative force. 



W d 



By the laws of variable forces vv = 2gfi=2g{\ -^J^- 



Taking the fluents, ''^^='^s(^~^o^ 5 ^"^ ^~'^'d ^ '^'^dx—x^. 



When x = d, v = i^'i.gd. And it appears that at the depth d 



the accelerative force is nothing; for then x-=-d, and I— -=0. 



A retarding force then commences, and the body will descend 

 till this retarding force destroys the whole of the velocity gene- 

 rated in moving through the space d. When the whole of the 

 velocity is destroyed the body will ascend, and after several vi- 

 brations will ultimately rest at the depth d. 



1 * 



Let 7 be the time of descent, then t-= — -=- = X 



The fluent may be found by means of a circular arc, but for 

 the present purpose it is perhaps better to express it in a series. 

 In that case we have 



. = 2^i^x(2+f^+^, + ^3 + &c.); or 



^^=4^-^x (2+f.+|£+4S3 + ^'0^- Hence it ap- 

 pears that the indentation increases very nearly as the square of 

 the time ; and that the proportion x : t^ which I have used is 

 sufficiently near the truth for practical purposes. 



By taking the value of d such that the resistance is exactly 



equal 



i 



