NOTICES AND ABSTEACTS 



OF 



MISCELLANEOrS COMMUNICATIONS TO THE SECTIONS. 



MATHEMATICS AND PHYSICS. 



Address by Professor P. G. Tait, M.A., F.E.S.E., President of the Section. 



In opening the proceedings of this Section my immediate predecessors have ex- 

 ercised their ingenuity in presenting its widely diifering component subjects from 

 their several points of view, and in endeavouring to coordinate them. What 

 they were obliged to leave unfinished, it would be absurd in me to attempt to com- 

 plete. It would be impossible, also, in the limits of a brief address to give a de- 

 tailed account of the recent progress of physical and mathematical knowledge. 

 Such a work can only be produced by separate instalments, each written by a spe- 

 cialist, such as the admiraole " Heports which form from time to time the most 

 valuable portions of our annual volume. 



I shall therefore confine my remarks in the main to those two subjects, one in 

 the mathematical, the other in the purely physical division of our work, which are 

 comparatively familiar to myself I wish, if possible, to induce, ere it be too late, 

 native mathematicians to pay much more attention than they have yet paid to 

 Hamilton's magnificent Calculus of Quaternions, and to call the particular notice 

 of physicists to our President's grand Principle of Dissipation of Energy. I think 

 that these are, at this moment, the most important because the most promising 

 parts of our field. 



If nothing more could be said for Quaternions than that they enable us to exhibit 

 in a singulai'ly compact and elegant form, whose meaning is obvious at a glance 

 on accoimt of the utter inartificiality of the method, results which in the ordinary 

 Cartesian coordinates are of the utmost complexity, a very powerful argument for 

 their use would be furnished. But it would be unjust to Quaternions to be con- 

 tent with such a statement ; for we are fuUy entitled to say that in all cases, even 

 in those to which the Cartesian methods seem specially adapted, they give as sim- 

 ple an expression as any other method ; while in the great majority of cases they 

 give a vastly simpler one. In the common methods a judicious choice of coor- 

 dinates is often of immense importance in simplifying an investigation ; in Qua- 

 ternions there is usually «o choice, for (except when they degrade to mere scalars) 

 they are in general utterly independent of any particular directions in space, and 

 select of themselves the most natural reference lines for each particvdar problem. 

 This is easily illustrated by the most elementary instances, such as the following : — 

 The general equation of Cones involves merely the direction of the vector of a point, 

 while that of Surfaces of Revolution is a relation between the lengths of that vector 

 and of its resolved part parallel to the axis ; and Quaternions enable us by a mere 



1871. 1 



