VOL. LXVII.] PHILOSOPHICAL TRANSACTIONS. 153 



having the same centre and radius as the equator de, to be carried about with 

 the revolving spheroid ; and suppose a sphere, whose radius is r, to revolve about 

 an axis pcir with the angular velocity e, and, while it is so revolving, let an ac- 



celerative force (p) equal to - X TrrT or — X _^ .-, according as b is less or 

 greater than r, urge the pole p, and the successive momentary poles as they be- 

 come such, to turn about a diameter of the contemporary mid circle in the 



manner expressed in art. 6, u being to t; as e + 



: r^- - b' cd ^^ r^ — b^ , c^ ^^ r- - b"- ^ cd ^ b> - r'- 



-- X zr-i:. to - X -rrj., or as e + - X ;:— ,- to - X 



b- e r"" + b^' ' e r^ + b- e r^ + b^' 



according as b is less or greater than r. Then will the motion of the surface of 

 this sphere be exactly the same as the motion of the said spherical surface carried 

 about with the revolving spheroid after receiving the impulse of the percussive 

 force. Therefore, having reference to our conclusions in the preceding articles^ 

 by substitution we readily obtain the solution to our problem. 



By substituting properly - X ^,. ^, or - X —7711 for f, we find, 



d^r dr r ^ -f b^ r x (cd -\- rp) 



V(d't- + 2crf/T + r^rO c- ^^^,„ _^ ^,., _^ ^,^, ^ <P '^ ^/(rf'e' 4: -cdri + r',') 



•3r* , f rr Qr^c 



;-> 3nd c + -7- = , , ,; . 



d'\ d r' + 6^ 



Which equations, respect being had to the conclusions in art. 8 and 9, indi- 

 cate that, whether b be less or greater than r, if an immoveable circle dl, whose 

 radius is = %, be conceived to be described in a plane uicli- 



c- 



ned to the plane of the equator of the spheroid (before the impulse) in an angle 



dr r^ -\- b^ 



whose sine (to the radius r) is = — X j^, so that the said cir- 



v'(4r* + (r= + b-'r X -) 

 c- 



cle touch the said equator in the point d in the section opvaE ; the spheroid 

 after the impulse will so revolve, that its equator will always touch and roll along 

 the said immoveable circle (dl,) the \elocity of the point of contact (along that 

 circle) being = ^ ,, , while the spheroid turns about its proper axis (oa) with 

 the primitive angular velocity c, and the poles o and a (by the said rolling of the 

 equator) describe circles, whose radii are each = — x '- 7-, pa- 



rallel to the said circle dl, with the angular velocity d (or their proper velocity 

 — ) which we supposed given to them by the impulse.* Thus the motion of 



r ' 



the spheroid consequent to the impulse appears to be remarkably regular. 



* Other ways of solving the problem are also suggested by the preceding article*.— -Orig. 

 VOL. XIV. X 



