I. On a Direct Method of estimating Velocities, Accelerations, and all similar 

 Quantities with respect to Axes moveable in any manner in Space, with Appli- 

 cations. By R. B. Haywakd, M.A. Fellow of St John's College, Reader in 

 Natural Philosophy in the University of Durham. 



[Read Fel. 25, 1856.] 



"...gardons-nous de croire qu'une science soit &ite quand on I'a reduite a des fonnules analytiques. Rien ne 

 nous dispense d'etudier les choses en elles-memes, et de nous bien rendre compte des idees qui font I'objet de nos 

 speculations." Poinsot. 



"...c'est une remarque que nous pouvons faire dans toutes nos recherches mathematiques ; ces quantites 

 auxUiaires, ces calculs longs et difiiciles ou Ton se trouve entraine, y sont presque toujours la preuve que notre esprit 

 n"a point, des le commencement, considere les choses en elles-memes et dune vue assez directe, puisqu'il nous faut 

 tant d'artifices et de detours pour y arriver ; tandis que tout s'abrege et se simplifie sitot 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 " Theorie de la Rotation " is so 

 Vol. X. Part I. 1 



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