VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 13 



^a?» + 5y2 + C«- = 1, 

 at the point, where the axis of angular velocity 01, whose equation is 



ai y as 



(Ol W2 ft)3 



meets it. Also reciprocally 01 is parallel to the normal to the ellipsoid, whose equation is 



x^ y' %^ 



at the point where OH meets it. 



Thus a simple geometrical construction enables us to determine 07, when O^ is given, 

 and vice versa. If now w be the angular velocity about 01, and / the moment of inertia 

 about the same line, the angular momentum about it must be Iw, since w is the total angular 

 velocity, and therefore the angular velocity about a line perpendicular to 0/is zero; hence 



lo) = h. cos HI, 

 an equation connecting h and w, the quantities / and HI being known when the above con- 

 struction has been made. 



24. If h be constant, and its direction OH invariable, it is plain from the above con- 

 struction that 01 will not in general remain fixed, nor m constant, for, by the motion of the 

 system about 01, the position of OH in the system is altered, and to this new position of OH 

 a new position of 01 will correspond, and then &> will change by reason of the variation of 



TT T 



. There is an exception however in the case where OH and 01 coincide, for then the 



rotation does not change the position of OH in the system : this can only be the case 

 when the radius 01 of the central ellipsoid is also a normal, that is, when it coincides with one 

 of the principal axes. Hence the principal axes are the only permanent axes of rotation of a 

 body acted on by no forces (as is implied in our supposition of h being constant) : in all 

 other cases the axis of rotation moves in the body and in space, and the angular velocity 

 about it varies. , 



25. If w be constant and its axis 01 fixed in the body, OH will also be fixed in the 

 body, and h will be constant ; but OH will then in general move in space, and the system 

 must therefore be acted on by forces, whose resultant couple has its axis perpendicular to OH 

 and in the plane of motion of OH. Hence the plane of the couple is HOI, if 01 be fixed in 

 space as well as in the body, and its moment is constant, since the velocity of OH is constant ; 

 thus the constraining couple on a body revolving uniformly about a fixed axis through its 

 centre of gravity is determined. 



In the exceptional case of a principal axis, OH is also fixed in space, and there is no 

 constraining couple. 



26. Before proceeding to the solution of the problem of a body's rotation about its 

 centre of gravity by a method more in accordance with the plan of this paper, it will be well 

 to shew how readily Euler's equations may be obtained from our principles. 



