150 PHILOSOPHICAL TRANSACTIONS. [aNNO 1777. 



be = k; and by substitution we get 



; = -^7— + r ^ = -r + 7' 1 being = - by the 



preceding article. Hence we find k = 



vi 1 fD f At 



-— ^--, — -T-; and it follows that -— r^- — ^, (= -) will be equal to theve- 



locity of the point a, and also of its opposite point (o) in the surface of the 

 sphere. It also follows, that k, the radius of eacii of the circles described by 

 those points, during the action of the force f, will be equal to 



rev 



By what is done it appears, that during the action of the force f, the motion 

 of the revolving sphere will be regulated by the circle p'p"p"' &c. on it (whose 

 radius is -— — -^-— - = ^-^—^^J continually touching and rolling along 



the immoveable circle p'p"f" &c. (whose radius is — — ^ = r-r so 



that the velocity of the point of contact be = v ■=- —. Considering the point 

 a as always urged from the points p', p", p", &c. and consequently its opposite 

 point (o) towards those points, it is necessary to observe, that according as u is 

 less or greater than e, the arc p'q (whose sine is — — ' ' ) will be less or 



greater than QO"; and the point (o) opposite to a, on the surface of the sphere, 

 will accordingly be at a greater or less distance than go° from p'. 



If M be negative, the arc p'r whose sine is ■ will be greater than go°. 



8. The motion of the sphere, according to the regulation in the preceding 

 article, is one motion compounded of the primitive motion of the sphere and the 



motion generated by the action of the force f. But conceiving — -, the 



velocity of the point q, to arise from an impulse given to it while the sphere 

 revolved about an axis of which a was an immoveable pole before such impulse, 

 and about which the mid-circle corresponding to that primitive axis revolved with 

 the angular velocity -7,- ^ ." , ,. .;* and considering that the force f, continu- 

 ally acting at right angles to the momentary direction of the point a and to the 

 plane of the said mid-circle, only serves to alter the position of the said primitive 

 axis; we may, by the help of what is done above, explain the motion which the 

 sphere will have, during the action of the force f, so as to retain in our ideas 

 the two primitive motions (one about the axis ao, and the other about a dia- 

 meter at right angles to that axis) as remaining distinct and unaltered. 



• Denoting this by c, and tlie velocity of « by </, ^(t^ + i/') is = c, agreeable to art. 1. — Orig. 



