1 88 Mr. Vince on the Motion of 
hence if we put PC, p = the circumference of a circle 
whofe radius is unity, then will pz = circumference deicribed 
a 1 
by the ooint P ; confequentiv — - ~ the number of revolutions 
J -*■ 1 2pz F 
required. 
Cor , If the (olid be a cylinder and r be the radius of its bafe. 
and therefore the number of revolutions = — - . 
M rB - 
proposition v.;. 
To find the nature of the curve deferibed by any point of a body, 
affected by frdtlion, when it def ends down any, inclined plane. 
Let efg (fig. 5-) be the body, the points a, r , r, as in Prop. I. 
and conceive j /, r n, to Be two indefinitely fmall fpaces deferibed 
bv the points s and r in the fame time, and which therefore 
will reprefen t the velocities of thofe points; but from Prop. I. 
the ratio of thefe velocities is exprefled by m x CB : a x CA, 
hence s t : r n :: m x CB : a x CA. With the center r let a 
circle vw be deferibed touching the plane LM which is parallel 
to AC at the point b, and let the, radius of this circle be fuch 
that, conceiving it to defeend upon the plane LM along with 
the body defeending on CA, the point b may be at reft, or the 
circle may roll without Aiding. To determine which radius, 
produce r s to x, parallel to which draw n dy , and produce nt 
to % ; now it is manifeft, that in order to anfwer the conditions 
above-mentioned, the velocity of the point x muft be to the 
velocity of the point r as 2 : i, that is, zx :yx.:i z: i, 
hence zy—yx-nr. Now zy : dt (:: ny :nd) :: r x : r s ; 
therefore dt = — x = ~ x nr, hence ts (~tdyds~tdynr~ 
r x ^ t x ~ 
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