Rotation of a Solid Body round a Fixed Point. 345 



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

 as R 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 ROTATION, 



WHICH BEING FIXED, THE AxiS WILL BE PERMANENT. 



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

 revolves with a rotation expressed by <o ; describe the ellipsoid 

 of gyration round the centre of gra- 

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

 The centrifugal force afrdm at any 

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

 components, wfpdm and o> 2 . R'P' . dm ; 

 r and p denoting the distances of the I 

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

 spectively ; the effect of the rotation 

 round R'R" 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 -;ufo 2 . OP. PR (vid. (8)); or, determining the point 

 R' by the condition OP . PR = OP' . PR', the centrifugal couple is 

 - /wo; 2 . OP'. P'R'. The resultant of the parallel forces is a force 

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

 to R'P', and equal to ^a> 2 . R'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 R'R" 

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

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

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

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

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



