148 PHILOSOPHICAL TRANSACTIONS. [aNNO 1777. 



caeteris paribus, the point of action with respect to the mid-circle (which point 

 we will now denote by (9) may be varied with a velocity greater or less than c; 

 and that, caeteris paribus, the velocity (v) of the momentary pole will be the 

 same, with what velocity soever (q) the point of action of the force f be varied; 

 the direction in which that force acts being always at right angles to the ray (Iq) 

 from the centre of the sphere, and to the tangent to the curve described by (q) 

 such point of action. 



Yet, though v continues the same whether, caeteris paribus, (u) the velocity 

 of the point q be greater, equal to, or less than e, the immoveable circle in 

 which the momentary pole will be found will not continue the same; that circle 

 being greater, equal to, or less than the circle whose radius is \/{r — q-), 

 according as u is less, equal to, or greater than e; as will be made more evident 

 by what follow: 



6. In fig. 16, let />', in the great circle Rp'aq'r, be one of the poles of the 

 axis about which the sphere rstv, whose radius is r, is revolving, according to 

 the letters \q's, with the angular velocity e, measured at the distance ?• from 

 the axis; and while it is so revolving let the said pole be urged to turn about a 

 diameeter of the mid-circle vq's towards q', by an accelerative force f ; and let 

 such force continue to act on the successive new poles p', p'", &c. as they 

 become such, always urging the sphere to turn about a diameter of the contem- 

 porary mid-circle, while the direction in which such perturbating force acts is 

 regulated in the following manner: 



Conceive the said revolving sphere to be surrounded by an immoveable concave 

 sphere of the same radius r. Then the momentary pole {p',p",p", &c,) will 

 always be found in some curve p'p'p'" &c. in the said concave sphere, and in 

 some curve p'p"p"' &c. on the revolving sphere; which last mentioned curve will 

 continually touch and roll along the other curve p'p"p"' &c. on the immoveable 

 .sphere, the force f and the direction in which it acts varying in any manner 

 whatever. LetF be invariable; then it is obvious, that the 2 curves so touching 

 each other will be circles; and if great circles p'q, f"q", f"'q'\ &c. be described 

 on the surface of the immoveable sphere whose planes shall be at right angles to 

 the plane of the circle p'p"p"' &c. the points q'q"q"' &c. in it, each 90° distant 

 from p'p'p'" &c. respectively, will be in a circle {q'q'q'" &c.) parallel to the said 

 circle p'p"p'" &c. Now as a regulation to the direction in which the force f 

 shall urge the momentary pole, let that direction be always a tangent to the 

 great circle so passing through that pole and the correspondent point q', q", or 

 q'', &c. while the arcs q'q", q'q", &c. are to the arcs p'p", p'p"', &c. respectively 

 in the constant ratio of u to v. 



The direction in which the force p acts being so regulated, it is obvious that 

 the radius of the circle p'p"p"' &c. being denoted by h, the radius ol the circle 



