12 mk r. b. hayward, on a direct method of estimating 



and h^ = 2m(y . v^ + w^y - w,jc - z .Vy + wjs - w^ss) 



= 1(my) . v^ - 2(m«) , v^ + S(m . y" + z') . w^ - '^(mxy)it)y - 2(m«*) . w^, 

 with similar expressions for Uy, u^ and hy, h^. 



From these equations it appears that, when the linear and angular velocities of the system 

 are referred to an arbitrary point 0, each depends in general on both the linear and the 

 angular momentum. If however be the centre of gravity, the linear velocity depends on 

 the linear momentum only, and the angular velocity on the angular momentum only, for 

 in this case "^{mx), "^(my), 2(m«) all vanish, and the equations become those, of which the 



types are 



u^ = 2(m) . Vj:, 



h^ = 2(TOy^ + z'')w^ - 1,(mxy) . Wy - '2imzx)a)^. 



22. Thus the motions of translation of the centre of gravity and of rotation about it are 

 independent, a property which is true of no other point. Also it is to be observed that 

 the direction of motion of the centre of gravity coincides with that of the linear momentum, 

 while that of the axis of angular velocity does not in general coincide with that of the angular 

 momentum. This is the cause of a greater complication in the problem of rotation than in 

 that of translation. In the former the passage from momentum to velocity involves the 

 changing of the direction of the axis as well as division by a quantity of the dimensions of 

 a moment of inertia, whose value depends on the position of the momental axis in the 

 system : in the latter the corresponding step involves simply division by a constant quantity, 

 the mass, without change of direction. If the operation by which the step is taken from 

 momentum to velocity, be considered as the measure of the inertia, we may express the above 

 by stating that the measure of the inertia of a system relatively to translation (the centre 

 of gravity* being the point of reference) is the mass of the system, and that the measure of 

 its inertia relatively to rotation is not a simple numerically expressible magnitude, but, in 

 Sir W. Hamilton's language, a quaternion, dependent on the position of the axis of angular 

 momentum or of that of angular velocity in the system. 



23. Confining our attention henceforth to the problem of rotation, we must first obtain a 

 more distinct idea of the relation between the axes of angular momentum and velocity. 

 We may obtain this from our previous equations for h„ hy, h., in their general form ; but 

 more simply when we consider our axes as coincident with the principal axes through 

 the centre of gravity. If ^, B, C denote the moments of inertia about these axes, 

 the equations become (substituting 1, 2, 3 as subscripts for x, y, z respectively) 



hi = Awi, h-z = Bb}2, A3 = Cw^ ; 

 hence the axis of angular momentum OH, whose equation is 



x y ss 

 hi hz hs' 

 is parallel to the normal to the central ellipsoid 



• It will be observed that, if the translation be referred to any other point than the centre of gravity, the measure of 

 inertia relatively to translation is also a quaternion. 



