118 On the Motion of Bodies affected hj Friction. 



the friction of the elementary parts of the base a, I, c, Sec. 

 will be as a x oC, b x bC, c x cC, &;c. also the effect of 

 the inertia of the corresponding parts of the bodv will be 

 as A X aC-, B x /'C-, C x cC-, Sec. Now when the 

 whole surface of the base and mass of the body are con- 

 centrated in P, the effect of the friction will be as a + i + c + 

 Sec. X CP, and of the inertia as A + B -t- C + Se c, x CP- ; 

 consequentlv a x aC -h I' x bC +c x cC + &cc. : a + b + c + 

 &c. x CP ':: A X flC- + B X bC"- + C X cC- + S:c. : 

 A4-B + C+ Sec. X CP^ ; and hence 



A X flC-- + B X /-C- H- C X cC- + &--C. X a-^-l-+ c-\- &c. 

 " X aC + b'xlC + c x7cT&c. X A + BT^T&c. 

 = (if S = the sum of the products of each particle into 

 the square of its distance from the axis of motion, T = 

 the sum of the products of each part of the base into its 

 distance from the center, 5 = the area of the base^ t = the 



solid content of the body) -; — . 



PROPOSITION IV. 



Given the velocity with which a body begins to revolve 

 about the center of its base, to determine the number of re- 

 volutions which the body wiJ make before all its motion be 

 destroyed. 



Let the friction, expressed by the velocity which it is able 

 to destroy in the body if it were projected in a right line 

 horizontally in one second, be determined by experiment, 

 and called F; and suppose the initial velocity of the center 

 of friction P about C to be a. Then conceiving the whole 

 surface of the base and mass of the body to be collected 



into the point- P, and (as has been proved in Prop. II.) -— 



will be the space which the body so concentrated will de- 

 scribe before all its motion be destroyed; hence if we put 

 X = PC, p = the circumference of a circle whose radius is 

 unity, then will pz =^ circumference described by the point 



P; consequently — - = the number of revolutions rc- 



\ 2pxF 



quired. 



Cor. If the solid be a cylinder and r be the radius of its 



base, then z = — , and therefore the number of revolutions 

 4 



2a2 

 3pr¥' 



PROPOSITION 



