438 Prof. Booth on the "Rotation 



end of the first instant the vertex of the axe of the resultant 

 moment will be found on the curve of double curvature the 

 intersection of the sphere and ellipsoid ; therefore u is con- 

 stant. 



XII. Now as u is constant, and/« also constant (l.^ftoo 

 is constant; but the angular velocity 



u = ^; (see (9.)) 



hence the angular velocity round the instantaneous axis of ro- 

 tation varies inversely as P. 



XIII. The angular velocity zr round the axis of the impressed 

 moment is constant during the motion. 



p 

 Let 6 be the angle between u and P, cos 6 = — , and 



ct = co cos 9 hence 



* f P f 

 vi = tj^- . — = -^-, a constant quantity . . . (10.) 



XIV. The magnitude of the centrifugal moment G varies as 

 the tangent of the angle between u and P. 



Resuming the equation (3.), G 2 = K w 2 — eo 4 D 2 , and put- 

 ting for K, w, and D their values given by (7.), (9.) and (6.) 

 and introducing the relation P 2 tan 2 6 = w 2 — P 2 , we find 



G = K^tan<9; 

 u 



f 

 or putting for — its value -ar, the angular velocity round the 



axe of the impressed moment, 



G = K*rtan0 (11.) 



XV. The curve which is the intersection of the ellipsoid and 

 sphere is on a cone of the second degree, whose circular sec- 

 tions are parallel to the central sections of the ellipsoid; let 

 the equations of the ellipsoid and sphere be respectively 



y/2 j/2 2 n 



^r + 4r + -V = j > and *' 2 +y 2 + *' 2 = *• 



