Professor Mac Cullagh's Lectures on Rotation. 145 



pulsive couple of a determinate magnitude and direction. The statical impulsive 

 couple thus conceived is the couple of principal moments. Let this couple be re- 

 presented by G, and act round the axisOR(fig. 1, p. 148); then the corresponding 

 axis of rotation will be the perpendicular OP, and the relation between G and 

 w may be thus found. Let the axes of co-ordinates be the axes of the ellipsoid 

 (4), the radius vector being determined by the angles (\, ;u, v), and the axis of 

 rotation by the angles (a, /3, 7). From mechanical considerations we obtain tlie 

 equations 



G cos X = Ap = fjLo) rt' cos a ; 



G cos fi = Bq = jxto Ir cos ^ ; 

 G cos V = Cr — fxu) c^ cos 7. 



Hence we obtain 



cos \ _a^ cos a cos n P cos /3 



(9) 



cos v c^ cos 7 ' cos V r cos 7 



G _ O cos 



The first two of these equations prove that the axis of rotation is the per- 

 pendicular on tangent plane of the elhpsoid, and the last equation gives the mag- 

 nitude of the rotation in terms of the impressed couple and quantities determined 

 by the natiu'e of the body itself Equations (9) are true, whatever be the forces 

 acting on the body ; if no forces act, G will be fixed in magnitude and posi- 

 tion in space, by the principle of conservation of areas, but will change its 

 position in the body, the axis of rotation accompanying it, and changing its 

 position both in the body and in space. 



VI. — Rotation produced by Centrifugal Force; pi'ii'ticular Properties of the Motion 



when no Forces act. 



The axis of rotation produced by the centrifugal couple always lies in the 

 plane of principal moments. This theorem may be thus proved : Let the 

 radius vector and perpendicular be drawn, which coincide with the axis of 

 principal moment and axis of rotation at any instant ; a line perpendicular to 

 the plane of radius vector and perpendicular is the axis of centrifugal couple ; 



VOL. XXII. u 



