288 = The Laws of Motion of a Cylinder, &e. 
pounded of the ratios of the depths penetrated, and 
of the fquare of the variable velocity; for, when 
one body flides along andther, the friction whjch 
continually retards its progrefs, arifes from the colli- 
fion of the fmall, but unavoidable, protuberances of 
the two touching furfaces. Now it is evident from 
the Laws of Motion, that the retarding force occa- 
fioned by the concuffion of two fuch protuberances, 
is directly as the velocity ; the number of fuch con- 
cuffions in a given time is in the fame ratio; and 
the number of protuberances acting together is pro- 
portional to the touching furface of the fliding body : 
Confequently the effect of friction is in a ratio com- 
pounded of the touching furface and the fquare of 
the velocity; that is, in the prefent cafe, as the depth 
penetrated drawn into the faid {quare.—From thefe 
confiderations it is obvious, that the following Quef- 
tion muft be refolved before a ‘Theory can be ob- 
tained from calculation, which will bear any analogy 
to the refult of experiment, 
Prostem IV. Every thing remaining as in the 
laft, excepting the variable part of the retarding 
force, which is now fuppofed to be as the depth 
penetrated drawn into the fquare of the velocity, it 
is required to determine the value of +, when 
‘o—v vanithes. 
Sotution. Let p, s, g, 0, mand x reprefent the 
fame quantities as in the laft Prog. alfo put f= 
the refiftance arifing from friction at the depth p, 
with the initial velocity ¢; then, fince the friction 
is 
