154 Rev. S. Haughton's Accou7U jf Professor Mac Cullagh's Lectures, ^c 



gravity O, and draw OP' parallel to E'R". The centrifugal force w-rdm at any 

 point (x, ij, z), may be resolved into two components, ^10.4. 



w-pdm Pnd w-.'R'Y.dm ; r and p denoting the distances 

 of the point from the axes R'R" and OP' respectively; 

 the effect of the rotation round R'R" is therefore the 

 same as an equal rotation round OP', together with a 

 number of parallel constant forces applied to each point 

 of the body. The rotation round OP' produces a cen- 

 trifugal couple represented by- /nwlOP.PR (vid. S); 

 or, determining the point R' by thecondition OP.PR = 

 OP'. P'R', the centrifugal couple is - /xw". OP'. P'R'. The resultant of the 

 parallel forces is a force applied at tlie centre of gravity, acting in the direction 

 parallel to R'P', and equal to fjno-. R'P'. Comparing this with the centri- 

 fugal couple, it is evident that the forces at O 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 permanent axis of rotation. The condition by which the point R' is 

 found is, that the triangle OR'P' is equal to the triangle ORP; hence, if an 

 ellipsoid confocal to the ellipsoid of gyration be described through the point R'^ 

 it will be perpendicular to the line R'R". The general construction for per- 

 manent axes is, therefore, the following. Let the ellipsoid of gyration be des- 

 cribed, and confocal ellipsoids ; any line which pierces one of these elhpsoids 

 at ritrht angles is a permanent axis of rotation for the point of intersection. 



