VOL. LXXX.] PHILOSOPHICAL TRANSACTIONS. 741 



perspicuous. I have therefore here taken its truth for granted, because it is also 

 exactly agreeable to the solutions of the other gentlemen, and saves the trouble 

 of repeating what they have done before. I have also shown wherein, and why, 

 his solution differs from theirs, and proved, as I think, undeniably, in what 

 respects it is defective. 



That the inertia, or, as M. Euler calls it, the momentum of inertia, is equal 

 to the fluent or sum of every particle of the body drawn into the square of its 

 distance from the axis of motion ; and the determination of the 3 permanent 

 axes, or the demonstration that there are, at least 3 such axes in every body, 

 round any one of which, if it revolved, the velocity would be for ever uniform, 

 I have also taken for granted, because these things have been proved before, and 

 all the gentlemen are agreed in them. Difficulties that occurred I have not con- 

 cealed, but shown how to obviate, and endeavoured to place the truth in as clear 

 a light as possible ; which to discover is my wish, or to welcome it by whom- 

 soever found. 



Mr. W. divides his work into several propositions, which he demonstrates in 

 an intricate algebraical method. 



Prop. 1 . While a globe, whose centre is at rest, revolves with a given velocity 

 about an axis passing through that centre, to find with what velocity any great 

 circle on the surface, but oblique to that axis, moves along itself. After the 

 determination of this velocity, Mr. W. infers these 2 corollaries. 



CoroL 1. Hence it follov/s, that in whatever manner a globe revolves, its 

 velocity, measured on the same great circle on its surface, must be the same at 

 the same time at every point of the periphery of that circle. — CoroL 2. Conse- 

 quently, however the plane of a great circle varies its motion, the velocity at 

 any instant is at every point of the periphery equal along its own plane. 



Prop. 2. Supposing the centre of a sphere to be at rest, while the surface 

 moves round it in any manner whatever ; then, if the same invariable point o, 

 considered as the pole of an axis of the sphere, be itself in motion, the angular 

 velocity of the spherical surface about that axis will be unequable, or that of 

 one point in it different from that of another. — Corollary. Hence, about what- 

 ever axis the angular motion of a sphere is equable, the pole of that axis, and 

 consequently the axis itself, must be at rest at the instant. Different motions 

 may have different correspondent poles, and consequently, when the motion is 

 variable, the place of the pole of equable motion on the surface may vary ; but 

 whatever point on the surface corresponds with that pole must at the instant be 

 at rest. 



Prop. 3. Let arc (fig. 6, pi. 8) be an octant of a spherical surface in motion, 

 while the centre is at rest ; and let the velocity of the great circle bc in its own 

 plane = a, and in a sense from b towards c ; that of ca in the sense from c 



