PHILOSOPHICAL TRANSACTIONS. [ANNO 1785. 



, . , „ a 2 (a x ra — vx «) 4 (a x ra — ti x rs) 1 . . 



velocity) a- : — :: — : J 2p x ml the space that could have 



been described from the time that the rotatory velocity was destroyed, till the pro- 

 gressive motion would have been destroyed had the friction continued to act ; 

 . a 1 (a x ra — v x rs) 1 2av x ra x rs — v 1 x rs* ,, , ., , , 



hence : = ; = the space described when 



2f 2f x ra 1 2f x ra 1 r 



the rotatory motion was all destroyed, hence 



(;\s 2 + 2rs x ra) x (a X ur — v x sr) 1 lav x ra x rs — v 1 x rs 1 , , . 



i -, : — -z j -, = whole space de- 



as 1 x ar 1 x 2f ' 2f x ra' " 



scribed by the body before its motion becomes uniform. 



Definition. — The centre of friction is that point in the base of a body on which 

 it revolves, into which if the whole surface of the base, and the mass of the body 

 were collected, and made to revolve about the centre of the base of the given 

 body, the angular velocity destroyed by its friction would be equal to the angular 

 velocity destroyed in the given body by its friction in the same time. 



Prop. in. — To find the centre of friction. — Let fgh (fig. 4) be the base of a 

 body revolving about its centre c, and suppose about a, b, c, &c. to be indefinitely 

 small parts of the base, and let a, b, c, &c. be the corresponding parts of the 

 solid, or the prismatic parts having a, b, c, &c. for their bases ; and p the centre 

 of friction. Now it is manifest, that the decrement of the angular velocity must 

 vary as the whole diminution of the momentum of rotation caused by the friction 

 directly, and as the whole momentum of rotation or effect of the inertia of all the 

 particles of the solid inversely ; the former being employed in diminishing the 

 angular velocity, and the latter in opposing that diminution by the endeavour of 

 the particles to persevere in their motion. Hence, if the effect of the friction 

 varies as the effect of the inertia, the decrements of the angular velocity in a given 

 time will be equal. Now as the quantity of friction (as has been proved from ex- 

 periments) does not depend on the velocity, the effect of the friction of the ele- 

 mentary parts of the base a, b, c, &c. will be as a X «c, b X be, c X cc, &c. also 

 the effect of the inertia of the corresponding parts of the body will be as a X ac 2 , 

 b X 6c 2 , c X cc'-, &c. Now when the whole surface of the base and mass of the 

 body are concentrated in p, the effect of the friction will be as (a + b -f c -f- &c.) 

 X cp, and of the inertia as (a + b + c -f &c.) X cp 2 ; consequently a X ac 

 -f b X be -\- c X cc + &c. : (a -f b + &c) X cp :: a X ac' 2 + b X be 2 -f c 

 X cc' 2 -}- &c. : (a + b + c + &c.) X cp' 2 ; and hence 



(A X AC 2 + B x Ac 2 + c x CC 5 + &C.) X (« + b + c + &c.) ... lt . 



CP = —. -—. , ; r— 5 — > . ; ; — r— = (it S = the Slim of 



(a x «c + o x 6c + c x cc + &c.) x (a + b + c + &c) v 



the products of each particle into the square of its distance from the axis of mo- 

 tion, t r= the sum of the products of each part of the base into its distance from 

 the centre, s = the area of the base, t = the solid content of the body) ——. 

 Prop, iv.— Given the velocity with ivhick a body begins to revolve about the 



