2 REPORT 1871. 



mark to separate the ideas of length and direction without introducing the cumbrous 

 and clumsy square roots of sums of squares -which are otherwise necessary. 



But, as it seems to me that mathematical methods should be specially valued in 

 this Section as regards their fitness for physical applications, -what can possibly 

 from that point of vieyv be more important than Hamilton's v? Physical ana- 

 logies have often been invoked to make intelligible va,rious mathematical processes. 

 Witness the case of Statical Electricity, -wherein Thomson has, by the analogy of 

 Heat-conduction, explained the meaning of various important theorems due to 

 Green, Gauss, and others ; and wherein Clerk-Maxwell has employed the proper- 

 ties of an imaginary incompressible liquid (devoid of inertia) to illustrate not 

 merely these theorems, but even Thomson's Electrical Images. [In fact he has 

 gone much further, having applied his analogy to the puzzHug combinations pre- 

 sented by Electrodynamics.] There can be little doubt that these comparisons 

 o-we their birth to the smaU intelligibility, per se, of -what has been called La- 



d^ d^ cP 

 place's Operator, t^ + "7-2 + X2) which appears alike in all theories of attraction at 



a distance, in the steady flow of heat in a conductor, and in the steady motion of in- 

 compressible fluids. But when we are taught to imderstand the operator itself we are 

 able to dispense -with these analogies, which, however valuable and beautiful, 

 have certainly to be used with extreme caution, as tending very often to confuse 

 and mislead. Now Laplace's operator is merely the negative of the square of Hamil- 

 ton's V, which is perfectly intelligible in itself and in all its combinations ; and 

 can be defined as gi-ring the vector-rate of most rapid increase of any scalar func- 

 tion to which it is applied — giving, for instance, the vector-force from a potential, 

 the heat-flux from a distribution of temperature, &c. Very simple functions of 

 the same operator give the rate of increase of a quantity in any assigned direction, 

 the condensation and elementary rotation produced by given displacements of the 

 parts of a system, &c. For instance, a very elementary application of v to the theory 

 of attraction enables us to put one of its fundamental principles in the foUo-wing 

 extremely suggestive form : — If the displacement or velocity of each particle of a 

 medium represent in magnitude and direction the electric force at that particle, 

 the con-esponding statical distribution of electricity is proportional evei-ywhere to 

 the condensation thus produced. Again, Green's celebrated theorem is at once 

 seen to be merely the well-kno\\-n equation of continuity expressed for a hetero- 

 geneous fluid, whose density at every point is propoi-tional to one electric potential, 

 and its displacement or velocity proportional to and in the direction of the electric 

 force due to another potential. But this is not the time to pm-sue such an inquiry, 

 for it would lead me at once to discussions as to the possible nature of electric 

 phenomena and of gravitation. I believe myself to be fully justified in saying 

 that, were the theoi-y of this operator thoroughly developed and generally known, the 

 whole mathematical treatment of such physical questions as those just mentioned 

 would undergo an immediate and enormous simplification ; and this, in its turn, 

 would be at once followed by a proportionately large extension of our knowledge*. 



* The following extracts from letters of Sir W. E. Hamilton have a perfectly general 

 application, so that I do not hesitate to publish them: — " De Morgan was the yery first 

 " person to notice the Quaternions in print ; namely in a Paper on Triple Algebra, in the 

 " Camb. Pliil. Trans, of 1844. It was. I think, about that time, or not very long after- 

 " wards, that he wrote to me, nearly as follows : — ' I suspect, Hamilton, that you have 

 " caught the right sow hy the ear ! ' JBetween us, dear Mr. Tait, I think that we shall begin 

 " the SHEADING of it ! ! " " You might without offence to me, consider that I abused the 

 " license of hope, which may be indulged to an inventor, if I were to confe.ss that I expect 

 "the Quaternions to supply, hereafter, not sneTelj mathematical methods, \>nt also phy- 

 " sical suggestions. And, in particular, you are quite welcome to smile if I say that it 

 " does not seem extravagant to me to suppose that a fiiU possession of those a priori prin- 

 " cipfes of mine, about the nviltiplication of vectors (including the Law of the Four Scales 

 " and the conception of the Extra-spatial Unit), which have as yet been not much more 

 " than hinted to the public, jiigut have led (I do not at all mean that in mi/ hands they 

 " ever woidd have done .so) to an Anticipation of the great discovery of Oersted." 



" It appears to me that one, and not the least, of the services which quaternions may be 

 " expected to do to mathematical analysis generally, is that their introduction -will compel 



