Rotation of a Solid Body round a Fixed Point. 345 



In (19) the positive or negative sign must be used according 

 as E is less or greater than the mean axis of the ellipsoid ; this 

 is evident from the composition of rotations, and from the con- 

 sideration that in the former case the axis of rotation falls inside 

 the cone (11), while in the latter case it falls outside. 



X. To FIND A POINT, IF ANY, IN A GIVEN Axis OF EOTATION, 



WHICH BEING FIXED, THE AxiS WILL BE PERMANENT. 



Let B'B" (fig. 4) be the given axis, round which the body 

 revolves with a rotation expressed by w ; describe the ellipsoid 

 of gyration round the centre of gra- 

 vity 0, and draw OP' parallel to B'B". 

 The centrifugal force uPrdm at any 

 point (#, y, z) may be resolved into two 

 components, u?pdm and <i> 2 . B'P' . dm ; 

 r and p denoting the distances of the 

 point from the axes B'B" and OP' re- 

 spectively ; the effect of the rotation 

 round B'B" is therefore the same as an 

 equal rotation round OP', together with 

 a number of parallel and equal forces applied to each point of 

 the body. The rotation round OP' produces a centrifugal couple 

 represented by - juw 2 . OP . PB (md. (8)); or, determining the point 

 B' by the condition OP . PB = OP' . P'B', the centrifugal couple is 

 - /W . OP' . P'B'. The resultant of the parallel forces is a force 

 applied at the centre of gravity, acting in the direction parallel 

 to B'P', and equal to juo> 2 . B'P'. Comparing this with the cen- 

 trifugal couple, it is evident that the forces at destroy each 

 other, and, therefore, the total result of the rotation round B'B" 

 is to produce a force acting at the point B', which has been just 

 determined. If this point be fixed, the axis B'B" will be a per- 

 manent axis of rotation. The condition by which the point B' 

 is found is, that the triangle OB'P' is equal to and in the same 

 plane with the triangle OBP : hence, if an ellipsoid confocal to 



