VOL. LXVIl.] I'HILOSOPHICAL TRANSACTIONS. 145 



DCE, from A towards h, with the velocity w. It is proposed to find the new 

 axis about which the sphere will revolve after receiving such impulse. 



Calling a/, parallel to DC, x; c/ will be = y^{r- — x^): the velocity of the 

 point a, about acb, before the impulse on a, will be = — ; and the velocity, 



about DCE, given to the same point a by the said impulse, will be = ^ . 



Which velocities of the point a being in contrary directions, if it be so situated 

 that they be equal, then, one destroying the other, that point will stop and 

 become one of the new poles sought, about which the former poles a and b will 

 revolve with the velocity w ; and the points d and e will revolve with the same 

 velocity c, as before the perturbating impulse on the point a ; but instead of 

 describing the great circle dgeh, their motion will be about the new axis ab; 

 about which they, as well as the points a and B, will describe lesser circles 

 parallel to the great circle de, in which the points d and e (de being at right 

 angles to ab) will revolve about the same axis ab, with the velocity v^ {c' + w'^). 

 Which being denoteil by e, and m and n being put for the sine and cosine of 

 the angle Aca to the radius 1, me will be = 7u, ne = c, and consequentlj 

 mne'' = cw. 



Now taking — = - — — —, in order to find that new axis ab, we have 



... rw rtv , 



from that equation x = ., ., . ,,,, = —- = ai. 



Further, it is obvious, that if a spheroid, a cylinder, or any other body, 

 whose centre of gravity is c, and proper axis acb, were, while revolving about 

 that axis with the same angular velocity c, to receive such an impulse as instantly 

 to give the point a the angular velocity 7u about dce; the axis about which that 

 spheroid, cylinder, or other body, immediately after the impulse, would revolve, 

 or would have a tendency to revolve, would be the same line ab. 



The great circle de (fig. 12), and any other great circle so situated with 

 respect to the axis of any revolving sphere, T shall denominate the mid-circle. 



2. In the manner above described, the poles of the sphere are, by the instan- 

 taneous impulse on the point a, instantly changed from a and b to a and b. 

 But if, instead of such impulse, a continued attractive force f, like that of gra- 

 vity, acted at a, fig. 14, and at the new poles a', a", &c. as they become such 

 by a successive change, caused by such continued action of the force f urging 

 the sphere at every instant to revolve about the diameter d'e', or d"e", &c. of 

 the contemporary mid-circle, the new pole, a', a", &c. would not instantly be at 

 a finite distance from the primitive pole a, but some finite time would be requi- 

 site, that by such successive change, the pole might be varied to a finite distance 

 from A : and the force f continuing invariable the velocity t; with which the pole 

 changed its place, would be expressed by - I being the time elapsed while the 



VOL. XIV. U 



