52 ' REPORT— 1903. 



and from very different points of view, developed a calculus of vectors by 

 defining their products. The applications of this new calculus to physics 

 (including dynamics) remained long restricted to a few of their followers. 

 It was, however, before their time that Faraday by his ' Lines of Force ' 

 and ' Fields of Forces ' gave a purely geometrical representation of the 

 phenomena of electricity and magnetism. Their analytical expression 

 requires vectors. The first who recognised this was Clark Maxwell, and 

 there can be little doubt that his success in putting Faraday's ideas into 

 analytical form was greatly due to his knowledge of quaternions. His 

 statements in the preface to, and in the preliminary chapter of, his ' Elec- 

 tricity and Magnetism ' are in this respect of great interest. I quote 

 from the latter : 'But for many purposes in physical reasoning, as dis- 

 tinguished from calculation, it is desirable to avoid explicitly introducing 

 the Cartesian co-ordinates, and to fix the mind at once on a point of space 

 instead of its three co-ordinates, and on the magnitude and direction of a 

 force instead of its three components. This mode of contemplating geo- 

 metrical and physical quantities is more primitive and more natural than 

 the other, although the ideas connected with it did not receive their full 

 development till Hamilton made the next great step in dealing with space 

 by the invention of his calculus of quaternions. 



' As the methods of Descartes are still the most familiar to students of 

 science, and as they are really the most useful for purposes of calculation, 

 we shall express all our results in the Cartesian form. I am convinced, 

 however, that the introduction of the ideas, as distinguished from the 

 operations and methods of quaternions, will be of great use to us in the 

 study of all parts of our subject, and especially in electro-dynamics, 

 where we have to deal with a number of physical quantities, the relations 

 of which to each other can be expressed far more simply by a few words 

 of Hamilton's than by the ordinary equations.' 



He goes on : ' One of the most important features of Hamilton's 

 method is the division of quantities of scalars and vectors.' 



I have heard these words quoted as a proof that Maxwell was alto- 

 gether in favour of Cartesian methods, and against quaternions and 

 vectors. But this is wrong so far as vectors are concerned. In fact, the 

 ideas which he took from Hamilton are chiefly t-wo—Jirst, vectors ; and 

 second, the classification of physical quantities into scalars and vectors. 

 It is well known that he attached vei-y great importance to the latter in 

 connection with the theory of ' Dimensions.' ' 



This classification has been carried further by Clifford. Certain 

 vector-quantities require position for their full specification ; Clifford says 

 such a quantity is 'localised,' and calls a localised vector a ' rotor.' - 

 Forces, spins, momentum, are examples. There are also localised scalars 

 like mass and energy. 



In connection with this subject the enforced absence, due to ill-health, 

 of Mr. Williams is much to be regretted. He has continued his valuable 

 work of the Theory of Dimensions, and has lately taken ' position ' into 

 account. It was hoped that he would communicate some of his recently 

 obtained results at this meeting, and thus bear witness to the importance 

 of vectors in this direction. 



' See his paper, 'Classification of Physical Quantities,' Proc. Lond. Math. Sor., 

 vol. iii. p. 224. 



- Professor Joly has pointed out to me that Hamilton has also considered these, 

 In his unpublished papers he calls them ' tractors.' 



