VOL. LXVII.J VHILOSOPHICAL TRANSACTIONS. 151 



In fig. 17, let ED be a great circle on the revolving sphere, of which q is a 

 pole, and let a smaller circle dl parallel to (mq) that which we have found will 

 be described by the point q, be drawn on the immoveable concave sphere, so as 

 to touch that great circle in the point (d) where the great circle qp'r cuts it; the 



radius of which lesser circle will be (=^(r'^ — k") =) -ttt-, — 2^' ^" /, — -^rr-, — ■\' 

 Then the revolving sphere, during the action of the force p, will so move, that 

 the first mentioned great circle (ed) shall continually touch and roll along the 

 said lesser circle dl, the velocity of the point of contact (along that circle) being 

 r ^u(e — «) jt and the sphere at the same time turnine; about the axis of 



which Q is a pole with the primitive angular velocity — _ J!" ^,. . 



Thus the primitive motion about the axis of which a is a pole is preserved 

 distinct, while that pole proceeds describing a circle, whose radius is 

 — ;r — ^■^, 7-; rr, With thc velocltv — TT rr which we supposed 



given to it. 



It is observable, that the last mentioned velocity will, according to this regula- 

 tion of the motion, be to the primitive angular velocity about the axis of which 

 (1 is a pole, as y to e — u, or as i; to m — e, according as u is less or greater than 

 e ; that is, according as the arc p'q is less or greater than gO°. 



9. From what has been said it follows, that denoting the two primitive angular 



velocities ^(/_^7)"+ v-) ^"^ ~:7~ -Zy + v) (specified in the preceding article) by 

 c and d respectively, the radius (fig. 16) of the drcXe. p' p" p'" &c. (or sine of the 

 arc p'q = p'q, &c.) will be = — ; the radius of the circle p'p"p"' &c. (or sine of 



the arc p'r = p"r, &c.) = ^^^^^q: gcrfrr + Hr ^) ' ^ S'"'^^* '^'''^'^ passing through 

 the primitive poles o and q, on the revolving sphere, will turn from the position 

 ORQ with the velocity 



^ measured at the mid-circle, or with the velocity — 7-7-^ ^ mea- 



sured at the fixed point r ; whilst those poles describe, with the velocity d, circles 

 parallel to p'p"p"' &c. the radius (k) of each of the circles (fig. 17) so described 



= V{d'c^^2cdrr + r^^y ^"^ ^^^ ^'^l^^^'ty _^_i^_ along the said circle dl 

 = c + -J : the upper or lower of the double signs taking place according as u 



* This is to the velocity of the point q as ^/{n. — k') to kj that is, as the radii of the arcs de- 

 scribed. — Orig. 



