VOL. LXVII.] PHILOSOPHICAL TKANSACTIONS. 14/ 



shall describe a conical surface. Then Im being denoted by r, and mo, per- 

 pendicular to In, hy q; -, the centrifugal force urging m in the direction om, 



will be to F, as r to v' (r'^ — q') = lo. Therefore f must be = — . 



Now while m is so revolving, if the force f ceases acting, the said particle m 

 will, it is obvious, immediately proceed to describe a great circle of the sphere 

 whose radius is r and centre /, of which great circle one of the poles will be 

 situated in a lesser circle parallel to, and Q0° distant from, that described by m 

 during such action of the said force; which pole, during such action, will change 

 its place in the said lesser circle in which it will at any time be found, with a 

 velocity v, which will be to e, as {s) the radius of the last-mentioned circle, to 

 q. But s will be = \/{r^ ~ ?'); therefore we have v. e:: v/(?" — q-) : q, and 

 ^Li J-L z= -. Consequently F = — ^ ^ will be = — X - = — , and v 



q e ^ •' rq r e r 



rF 



e 

 Let now 711 be a point on the surface of a sphere whose centre is /, and radius 



Im = r; and let the sphere revolve about an axis so that m shall describe a great 

 circle with the velocity e. If then such a motive force begins to act on the 

 sphere, that, continuing its action, the point 7n shall always be urged by the 

 invariable accelerative force f, to move in a direction at right angles to the ray 

 Im, and to the tangent to the curve which m will describe; that point it is 

 obvious will, in consequence of the action of that force, describe a lesser circle 

 of the same radius (9) as that described by the particle m, when fastened to a 

 string, and acted on by the force p as abovementioned; and the centre of the 

 sphere being always considered as at rest, one of the momentary poles of the 

 sphere will describe a circle whose radius will be = v/(r'^ — q'^) parallel to, and 

 90° distant from, that described by the point ?«. For if the said force were to 

 cease acting, that point of the sphere would describe a great circle, as would the 

 particle m at the string in the like case; and therefore both the said particle and 

 the point m of the sphere, at every instant having the same tendency, and being 

 acted on by equal accelerative forces, the effect will be the same with respect to 

 the motion of each. Consequently, v being put to denote the velocity with 

 which the momentary pole changes its place, in the circle which it will describe 

 while the motive force producing the accelerative force f acts on m as just now 



mentioned, v will be = — , the same as in the preceding article, e here de- 

 noting that velocity which we there denoted by c. 



5. Referring the point of action of the perturbating force to the midcircle, 

 we have not hitherto considered that point as varied with a greater or less velo- 

 city than (e) that of the point m; that is, with reference to such circle we have 

 always considered the point m as the point of action. But it is obvious that, 



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