524 



" The forces resulting from the centrifugal forces of a body of any 

 figure, revolving round an axis passing through a fixed point O, may 

 be found elegantly by the ellipsoid of which we have already made 

 so much use. Let a plane at right angles to the given axis OK, 

 and cutting it in K, touch the ellipsoid 

 in I ; the centrifugal forces will be re- 

 duced to a pair whose moment is OK X 

 KI X «"* (" being the angular velocity), 

 whose plane is OKI, and direction as 

 marked in the figure ; and a single force 

 equal to MjOw", p being the distance of 

 the centre of gravity from OK. 



« If OK pass through thecentre of gravity, there remains only the 

 pair. The perpendicular OK is the radius of gyration for the axis 

 OK. Particular consequences of these things are numerous. 



" Any line, taken at random in a body, may not be a principal axis. 

 All the principal axes parallel to a given hne lie in the same plane; 

 and the points of their lengths which must be fixed, in order that they 

 may be principal axes, will lie in a hyperbola. Suppose in this case 

 the point O (preceding fig.) to be the centre of gravity, and OK to 

 be parallel to the given line, and describe through I an equilateral 

 hyperbola whose asymptotes are OK and OL; then all the 

 principal axes, as O'K', 0"K", parallel to OK, lie in the plane 

 OKI, and the points O', O" of their lengths, which must be fixed, 

 are at their intersections with the hyperbola. 



" By means of theorems of this nature, all of which are proved 

 geometrically, without any calculation, I have been able to give a 

 complete geometrical solution of the problem of the motion of a solid 

 body not acted on by any forces. If it be acted on by given forces, 

 the differential equations in A, B, C and jo, y, r, which are given by 

 all the writers on mechanics, are direct consequences of the first 

 principles, without the intervention of any calculation. 



<« Another thing with which I had occupied myself is the attrac- 



* Multiplied by the mass of the body, or by M. 



