VOL. L.] PHILOSOPHICAL THANSACTIONS. l/l 



rotation of the whole sphere, or of all the particles, be supposed, in proportion 

 to the momentum of an equal number of particles, revolving at the distance oa 

 of the remotest point a, as n is to unity. 



It is well known, that the centripetal force, by which any body is made to 

 revolve in the circumference of a circle, is such as is sufficient to generate all the 

 motion in the body, in a time equal to that in which the body describes an arch 

 of the circumference equal in length to the radius. Therefore, if we here take 

 the arch ar = oa, and assume m to express the time in which that aich would 

 be uniformly described by the point a, the motion of a particle of matter at a 

 (whose central force is represented by/") will be equal to that which might be 

 uniformly generated by the force^^ in the time m ; and. the motion of as many 

 particles (revolving all at the same distance) as are expressed by en (which by hy- 

 pothesis is equal to the momentum of the whole body), will consequently be 

 equal to the momentum that might be generated by the force/ X en, in the 

 same time m. Whence it appears, that the momentum of the whole body 

 about its axe p/}, is in proportion to the momentum generated in a given particle 

 of time m\ by the given force f in the direction al, as ncf X m is to f X m, or, 



as unity to — -p X ~ (because the quantities of motion produced by unequal 

 forces, in unequal times, are in the ratio of the forces and of the times conjunctly). 

 Let therefore al be taken in proportion to am, as -^. X — is to unity (supposing 

 AM to be a tangent to the circle abcd in a, and let the parallelogram amnl be 

 compleated ; drawing also the diagonal an ; then, by the composition of forces, 

 the angle nam (whose tangent to the radius oa is expressed by oa X —.. X — ) 

 will be the change of the direction of the rotation, at the end of the aforesaid 

 time m\ But this angle being exceedingly small, the tangent may be taken to 

 represent the measure of the angle itself; and if z be assumed to represent the 

 arch described by a, in the same time in about the centre o, we shall also have 

 — = — = — , and consequently oax-7.X- = zx-7-. Whence it ap- 

 pears, that the angle expressing the change of the direction of the rotation, 

 during any small particle of time, will be in proportion to the angle described 



about the axe of rotation in the same time, as — is to unity 



ncf 



G. E. I. 



Though in the preceding proposition the body is supposed to be a perfect 

 sphere, yet the solution holds equally true in every other species of figures, as is 

 manifest from the investigation. It is true indeed, that the value of n will not 

 be the same in these cases, even supposing those of c, /"and f to remain un- 

 changed ; except in. the spheroid only, where, as well as in the sphere, n will be 

 = \\ the momentum of any spheroid about its axis being two-fifths of the mo 



z 2 



