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I.

On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space, with Applications. By R. B. HAYWARD, M.A. Fellow of St John's College, Reader in Natural Philosophy in the University of Durham.

[Read Feb. 25, 1856.]

"...gardons-nous de croire qu'une science soit faite quand on l'a réduite à des formules analytiques. Rien ne nous dispense d'étudier les choses en elles-mêmes, et de nous bien rendre compte des idées qui font l'objet de nos spéculations." POINSOT.

"...c'est une remarque que nous pouvons faire dans toutes nos recherches mathématiques; ces quantités auxiliaires, ces calculs longs et difficiles où l'on se trouve entraîné, y sont presque toujours la preuve que notre esprit n'a point, dès le commencement, considéré les choses en elles-mêmes et d'une vue assez directe, puisqu'il nous faut tant d'artifices et de détours pour y arriver; tandis que tout s'abrége et se simplifie sitôt qu'on se place au vrai point de vue." Ibid.

THE general principles, which I have endeavoured to keep in view in the investigations of this paper, are those contained in the above quotations from Poinsot. My object is not so much to obtain new results, as to regard old ones from a point of view which renders all our equations directly significant, and to develop a corresponding method, by which these equations result directly from one central principle instead of being (as is commonly the case) deduced by long processes of transformation and elimination from certain fundamental equations, in which that principle has been embodied.

The frequent occurrence of exactly corresponding equations, (though this correspondence is sometimes disguised under a different mode of expression) in many investigations of Kinematics and Dynamics suggests the inquiry whether they do not result from some common principle, from which they may be deduced once for all. An investigation based on this idea forms the first part of this paper, in which it will be shewn how the variations of any magnitude, which is capable of representation by a line of definite length in a definite direction and is subject to the parallelogrammic law of combination, may be simply and directly estimated relatively to any axes whatever. The second part is devoted to the general problem of the dynamics of a material system, treated in that form which the previous Calculus suggests, together with a development of the solution in the case of a body of invariable form.

Since whatever novelty of view is contained in this paper consists rather in the relation of the details to the general method than in the details themselves, much that is familiar to every student of Dynamics must be repeated in its proper place, but it is hoped that such repetition will in general be compensated by a new or fuller significance being obtained. As regards the problem of rotation, M. Poinsot's solution in the "Théorie de la Rotation" is so VOL. X. PART I. 1

: B: HAYWARD, ON A DIRECT METHOD of estimatTING complete and so entirely satisfies the conditions expressed in our quotations above from that work, as to leave nothing to be desired. But it does not appear to me that his method, which depends essentially on the summation of the centrifugal forces, is so widely applicable beyond the limits of this particular problem as that by which the same results are obtained in this paper: but be this as it may, any new point of view, if a true one ("vrai point de vue") has its special advantages, and on this ground may claim some attention.

1.

SECTION I.

The Method, with some kinematical Applications.

As we shall here be concerned only with the directions of lines in space, and not with their absolute positions, it will be convenient to suppose them all to pass through a common origin O, and to define the inclination of two lines as OP, OQ by the arc PQ of the great circle, in which the plane POQ meets a sphere whose centre is 0 and radius constant. We shall also suppose any linear velocity, acceleration or force, represented by a length along OP, to tend from O towards P, and any angular velocity or the like, represented in like manner, to tend in such a direction about OP that, if OP were directed to the north pole, the direction of rotation would coincide with that of the diurnal motion of the heavens.

2. Let u denote any magnitude, which can be completely represented by a certain length along the line OU, and which can be combined with a similar magnitude v along OV by means of a parallelogram, like the parallelogram of forces or velocities. Then of course u may be resolved in different directions by the same principles, and thus if we adopt rectangular resolution, the resolved part of u along OP will be u cos UP, which may be denoted by up. We proceed to inquire how up varies by a change in the position of OP.

3. Suppose OP to be a line moving in any manner about O, and that it shifts from OP to a consecutive position OP in the time dt; and conceive that this motion arises from an angular velocity about an instantaneous axis OI. Resolve into its components cos 1U about OU and 2 sin IU about a line in the plane IOU, perpendicular to OU: and farther resolve this latter component in the plane perpendicular to OU into the components Q sin IU .cos IUP in the plane POU and § sin IU. sin IUP perpendicular to the same plane.

Then the component in OU and that perpendicular to it in the plane POU produce displacements of P perpendicular to the arc UP, and consequently do not ultimately alter the length of the arc UP, so that Up remains ultimately unchanged so far as the motion of OP is due to these components: but the component perpendicular to the plane POU increases UP by the arc sin IU. sin IUP. dt, and therefore the increment of u, from this component (being equal to u. sin UP. d. UP) is

But the other increments being zero, this is the total increment of up, wherefore we have