334 Rotation of a Solid Body round a Fixed Point. 



taking more particular axes of co-ordinates. Let the axis of 



rotation be the axis OZ, and the plane of radius vector and 



perpendicular be the co-ordinate plane 



XOZ. In the accompanying figure OB' 



and OP' are the radius vector and per- 



pendicular of the ellipsoid (2), and OB, 



OP the radius vector and perpendicular 



of the ellipsoid (4), which is reciprocal 



to the former; the rotation is positive, 



in the direction indicated by the arrow. 



As the rotation is round the axis of z, it is Fi s- i- 



easy to see that the statical couple produced by centrifugal force 



will have for components, round the axes of x and y respectively, 



the quantities w^yzdm, ta^xzdm taken with their proper sign; 



i. e. the components are o> 2 i', w W ; J7, M' being coefficients 



in the equation of the ellipsoid 



A a? + BY + C'z* - 2L'yz - 2M'xz - IN'xy = /u. 



The tangent plane to this ellipsoid, applied at the point (x, y, s), 

 will be 



(Ax - M'z - N'y) x + (B'y - N'x - L'z] y f + (C'z - L'y - M'x] z' = p. 



At the point B' the tangent plane will be perpendicular to the 

 plane XOZ, and will be found by making x = 0, y = 0, and de- 

 stroying the coefficient of y' in the preceding equation. These 

 conditions give us L' = 0, which proves that the statical couple 

 produced by centrifugal force lies altogether in the plane XOZ. 

 The equation of the tangent plane is the same as the equation 

 of the line B'P', and is 



Hence we obtain 



M' 



The value of the centrifugal couple is wW, which is found 



