1.46 PHILOSOPHICAL TRANSACTIONS. [aNNO 1777. 



pole is varying from a to a, and z the length of the arc aa. Therefore the 

 velocity with which the pole will change its place, during such action of the 

 force p, will be expressed in the same manner as the velocity v, of a body moving 

 uniformly from a to a in the time t may be expressed; that is, in both cases v 

 will be = -. But there is a material difference between the motion of a body 

 so moving from a to a, and the change of place of the pole a, d', &c. the former 

 is permanent, and will continue to carry the body forward without the action of 

 any force whatever; whereas the latter will instantly cease, and the axis will keep 

 its position, if the force f ceases to act on it; like as the varying direction of a 

 projectile near the earth's surface would immediately cease to change, if the force 

 of gravity ceased to act. 



It is observable, that while the force f acts, and the revolving sphere, in con- 

 sequence of such action, every moment takes a new axis, the angular motion about 

 the axis will continue invariable; the action of such force only altering the axis 

 without altering the angular velocity of the sphere about it: like as the direction of a 

 moving body is altered, without altering its velocity, by an attractive force continu- 

 nually actingon it, in a direction at rightangles to that in which the body is moving. 

 And if ever the force p shall cease to act, the sphere will instantly revolve with its 

 primitive velocity c about the axis it then may have been brought to take by the 

 preaction of that force. The new axis, about which the sphere has such ten- 

 dency to revolve, at any instant during the action of the force f, I shall call the 

 momentary axis; and its poles the momentary poles. 



3. From the equation — = ^^ (art. 1) we nave - = — -. Now 



if a continued attractive force (f) act during the time t as abovementioned, instead 

 of the instantaneous percussive force at a, according to the doctrine of fluxions 

 we must, instead of 2i', take j^-, or its equal f ^, and .r instead of a:, in the expres- 

 sion -; therefore, in this case we have ^ = -^ = ,, ,, ■- — ;r. Whence, put- 



X X X V(' — t) ' l^ 



ting z for the arc (aq', or ao", &c.) whose sine is x, and writing z for its equal 



— — , we eet ^ = c, or i; = — . Hence, v denoting the velocity with 



which the momentary pole {a', a", &c.) changes its place, during the action of the 

 accelerative force f, we have z = vt ■= — , and consequently v =■ —. 



4. The value of v may also be determined in the following manner (fig. 15). 

 Conceive a very thin string, without weight, to have one of its ends fastened 

 to a fixed point/, and the other to a heavy particle of matter w; also conceive 

 such particle so to revolve with the velocity e, about the line in, that a certain 

 accelerative force f (like that of gravity referred to a certain direction) con- 

 tinually acting on the said particle m, in a direction at right angles both to the 

 string Im, and to the tangent to the curve in which in is moving, the string 



