436 Prof. Booth on the Rotation 



Let D be the moment of inertia round a right line passing 

 through the centre, making the angles \, ft, v with the axes 

 of coordinates, then 



D = A cos 2 A + B cos 2 ft + C cos 9 v (5.) 



Poisson, Traite de Mecanique, torn. ii. p. 56. 

 Or, putting for A, B, C their values given by (4.), 

 D = ?i 3 (a 2 cos 2 X + 6 2 cos 2 p + J 2 cos 2 v). 



Now the part within the brackets is the square of a per- 

 pendicular from the centre of the ellipsoid on a tangent plane, 

 making the angles X, ft., v with the axes, calling this perpendi- 

 cular P, we have 



D = n 3 P 2 (6.) 



IX. Assume the impressed moment 



K m n*fu, . \ (7.) 



u being the central semidiameter of the ellipsoid perpendicular 

 to the plane of the impressed moment K. The product/ft is 

 of course constant, it will be shown presently that f and u 

 are each constant. 



X. The axis of instantaneous rotation coincides iscith the 

 perpendicular from the centre on the tangent plane to the ellip- 

 soid, drawn through the vertex of the semidiameter u, and the 

 angular velocity round this axis is inversely proportional to 

 this perpendicular. 



Let p, q, r be the angular velocities round the three prin- 

 cipal axes, «, /3, y the angles which the axis u of the im- 

 pressed moment makes with the same axes, x,y,z the co- 

 ordinates of the extremity of w, then by (III.) we have 



K cos a „ „ _ x . o fl , 



p = — r — , now K = ?rfu, cos a — — , A =« d a% hence 



A. u 



fx . ,.. 

 p = "—- ; in like manner 

 a 1 



? = #. »■ = £ < 8 -> 



Hence P * +?+,» = «?=/> ||1 + |1+ -£} = £ (9.) 



But the cosines which the axis of instantaneous rotation makes 



7) o r 

 with the axes of coordinates, are — , — , — ; and from the 



CO CO CO 



values of p, q, r just found, we get 



p Px q Py r Pz 



17 ~ ~' ~<o ~~ 6* ' "c7 "" ~W' 



