1 6 Professoi' Booth on the Rotation 



Let w' be the angular velocity round this axe, p', q', r' its 

 components round the axes of coordhiates ; then as the an- 

 gular velocity round any principal axe is equal to the impress- 

 ed moment resolved perpendicularly to this axe divided by 

 the corresponding moment of inertia, we find 



p' = G~- =f~ tan . ^ from (3.) and (4.). 



A a^ 



Now by (15.), -^ = -^/tan 6, hence p' =f-jf' 



In like manner, <?' =/^, ?•' =/-^. . . . (35.) 



But the cosines of the angles which the instantaneous axis of 

 rotation due to the centrifugal force makes with the axes of 



coordinates are ^, -^, — ; ; and the cosines of the angles 



O) O) 0) 



CC XI ^ 



which « makes with the same axes being — , ■^, — , we shall 



° u u u 



have for the cosine of & the angle between the axis of the 

 impressed moment, and the axis of rotation due to the centri- 

 fugal moment, the formula 



cos 



a _ LfJL ^A-l. ^A.± jlf "I 

 u ^ a2 dt'^b'^ dt^ c^ dt J' 



Now the part within the brackets is the differential of the 

 equation of the ellipsoid, and therefore = 0; hence 



cos 6' = 0, or 6' =90. 



XXVII. To find the component of the angular velocity due 

 to the centrifugal force, resolved along the axis of instan- 

 taneous rotation. 



Let 8 be the angle between the axes of the rotation caused 

 by the impressed moment and the centrifugal force, then 



^ J)' Vx d Fy r Vz , ,^ , 



cos b = -r . — 2" + ~-r --^- + -p — g- (see art. A.) ; 



or putting for;/, 17', ?' their values given by (35.), 



I 5. ^ ^ [ ^ dx y dy z dz\ 

 a,'cos8 = P/|^^+^^ +^^> 



Now the part within the brackets is equivalent to 



