This part, which, upon account of physical difficulties, is indeed somewhat slippery and perplexing, I shall make the principal subject of this essay. GENERAL PROPOSITION. _Supposing an homogeneous sphere_ OABCD (Fig. 1.) _revolving uniformly about its centre, to be acted on at the extremity_ A _of the radius_ OA, _in a direction_ AL _perpendicular to the plane of the equator_ ABCD, _and parallel to the axis of rotation_ Pp, _by a given force, tending to generate a new motion of rotation at right angles to the former; It is proposed to determine the change, that will arise in the direction of the rotation in consequence of the said force._ [Illustration: FIG. 1.] Let _F_ denote the given force, whereby the motion about the axis P_p_ is disturbed, supposing _f_ to represent the centrifugal force of a small particle of matter in the circumference of the equator, arising from the sphere’s rotation; and let the whole number of such particles, or the content of the sphere, be denoted by _c_: let also the momentum of 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, whereby 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_, wherein 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 arch would be uniformly described by the point A, the _motion_ of a particle of matter at A (whose central force is represented by _f_) will be equal to _that_, which might be uniformly generated by the force _f_, in the time _m_; and the motion of as many particles (revolving, all, at the same distance) as are expressed by _cn_ (which, by hypothesis, is equal to the momentum of the whole body), will, consequently, be equal to the momentum, that might be generated by the force _f_ × _cn_, in the same time _m_. Whence it appears, that the momentum of the whole body about its axe P_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_ × _m_ is to _F_ × _m’_, or, as _unity_ to (_F_/_ncf_) × (_m’_/_m_) (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 (_F_/_ncf_) × (_m’_/_m_) 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 × (_F_/_ncf_) × (_m’_/_m_)) will be the change of the direction