June 15, 1893] 



NATURE 



149 



which is a directed quantity." Unfortunately for this argument 

 V does not denote the velocity in its complete conception— it 

 simply measures the speed. The physicist may think of velocity 

 as being a vector quantity ; but in ordinary analysis the vector 

 is not symbolised. We deal only with tensors and scalars. It 

 would be well, I think, if the strict meaning of vector were 

 clearly borne in mind. A vector Is a directed line in space, and 

 may be used to symbolise all physical quantities which can be 

 compounded according to the well-known parallelogram law. 

 Displacement is perhaps the simplest conception that can be so 

 symbolised. Velocities, concurrent forces, couples, &c., are 

 in the same sense vector quantities. Now it can be proved 

 rigorously that quadrantal versors are compounded according to 

 this very addition law. On what grounds, then, are they re- 

 fused admittance to the order of vectors ? If a vector cannot be 

 a versor in product combinations, what is the significance of the 

 equation ij = k? Regarding this Dr. Macfarlane vouchsafes 

 no remark, save that it is possible to get along without its use. 

 As he himself has not done so, such a possibility lies altogether 

 outside our consideration. Again, I fail to see what "physical 

 considerations" have to do with mathematics of the fourth 

 dimension. 



Dr. Macfarlane says that the "onus probandi lies on the 

 minus men." To my mind there is no question of proof at all. 

 That the unit vector a should fulfil the equation a' = + I is 

 a bare assertion on the part of Dr. Macfarlane and Mr. 

 Heaviside supported by such words as "natural, simple, con- 

 ventional," and the like. The equation a' = -1- I is a pure 

 assumption, having no better physical basis than the assump- 

 tion that a" = - I. But in quaternions this is not the assump- 

 tion. The assumption is— as Dr. Macfarlane admits— that pro- 

 ducts are to be associative. Hamilton, in fact, invented his 

 calculus so as to have its rules differing as little as possible from 

 the recognised rules of algebra. The commutative law had to 

 «o, but the others were kept (see Preface to Lectures, §§ 50—56). 

 Ill the system he advocates. Dr. Macfarlane loses the associative 

 principle, and— as I think I show in my paper— gains nothing 

 but a positive sign and an undesirable complexity in trans- 

 forming by permutations. 



As a calculus, quaternions may be developed quite as readily 

 from the conception of the product as from that of the quotient. 

 But in my paper I was arguing against Prof. Gibbs's dictum that 

 the quaternion as a quantity corresponded to nothing funda- 

 mental in geometry. The extremely simple geometrical con- 

 ception of a quaternion as a quotient of two vectors sufficiently 

 meets Dr. Macfarlane's query, " Is not the product always the 

 simpler idea?" It is certain that the quotient of any two 

 like quantities has always a meaning; the product is often 

 meaningless. 



In the particular geometrical development of quaternions 

 which I indicate in my paper, it can be shown that the quater- 

 nion, originally defined as the quotient of two vectors, can also 

 be represented as the product (Dr. Macfarlane inadvertently 

 misquotes "quotient") of two quadrantal versors, and this 

 quite independent of the truth that quadrantal versors obey the 

 vector law. 



Dr. Macfarlane evidently grudges Prof. Tait (properly, 

 Kelvin and Tait) the use of any but quaternion symbolism. 

 Of course, when v'-'z' occurs in ordinary non-quateinion analysis, 

 it is used in the sense of the tensor, for only as such can it 

 come in. This surely hardly needed to be pointed out. In 

 quaternions there is no doubt whatever that v(V») = (VV)a» 

 = v'a ; and therein, as in all the higher physical applications, 

 ihe flexibility and power of Hamilton's calculus are at once 

 apparent. 



In conclusion, let me say that no reasonable man can possibly 

 object to investigators using any innovations in analysis they 

 may find useful. But in the present case there is a very serious 

 objection to the innovators condemning the system, from which 

 they have one and all drawn inspiration, as " unnatural " £.nd 

 "weak," without in any way showing it so to be. That they 

 can re-cast many quaternion investigations into their own mould 

 does not prove their mould to be superior or even comparable to 

 the original. Yet, in so far as they possess much in common 

 with quaternions, the modified systems used by Gibbs, Heavi- 

 side, and Macfarlane cannot fail to have many virtues. 



*' His form had not yet lost 

 All her original brightness, nor appeared 

 Less than Archangel ruined." 



Edinburgh University, May 29. 

 NO. 1233, VOL. 48] 



C. G. Knott. 



The Fundamental Axioms of Dynamics. 



My reasons for holding that the fact that potential energy 

 belongs to a system rather than a particle is hostile to the idea 

 of the idenliiy of eneigy, are briefly these. If two pieces of 

 kinetic energy, are successively transformed and added to a 

 system as potential energy, and then some of the potential 

 energy is retransformed into kinetic, we cannot say w/iiVAofthe 

 original kinetic energies thus makes its reappearance; for while 

 both weie potential they had no local habitation within the 

 system, and so could not be distinguished from each other. 



The objections to including the ether as one of the " bodies " 

 between which contact actions occur, without further explana- 

 tions, are admirably stated by Prof. Rucker ; but I should like to 

 go even further than he does, and point out that if " contact ac- 

 tion "means, nly "action at constant distance" it has not yet been 

 shown how, by such action, kinetic energy comes to be trans- 

 ferred from one body to another. For if the bodies " move 

 over the same distance," and have at any moment the same 

 velocity, their kinetic energies are both inci eased or decreased 

 together ; whereas what we wish to show is how that of the one 

 body may increase while that of the other decreases, and why 

 the increase in the one case is equal to the decrease in the other. 

 For example, it may be that in a perfect fluid such transference 

 of kinetic energy actually takes place ; but the question is, has 

 Prof. Lodge explained this as a case of " contact action " or 

 "action at constant distance"? What are the things or 

 " bodies " which in this case are actually in contact, and which 

 move over equal distances while the action is going on ? Or 

 between what points is the "constant distance " to be measured ? 

 Prof. Lodge has not shown in his last paper or in those in the 

 Phil. Mag. how "potential energy" can be explained by con- 

 tact action, nor how kinetic energy can be transferred by 

 contact action alone. But perhaps the answers to these questions 

 are included in the "something more definite" which Prof. 

 Lodge now realises that he has " to say concerning the functions 

 of the ether as regards stress " ? 



The third paragraph of Prof. Lodge's letter is evidently a 

 joke. I certainly suppose that the denial of action at a distance 

 means that material particles are without direct influence upon 

 one another until they touch ; i.e. that any influence they do 

 exert is indirect, and takes place through their both touching 

 something else. Indeed I indicated this in my last letter ; but 

 Prof. Lodge apparently hoped I would overlook his omission of 

 the word "direct," and that so the joke would go against me ! 



Edward T. Dixon. 



Trinity College, Cambridge, June 10. 



Chemical Change. 



In the current number of the Proceedings of the Chemical 

 Society, Prof. H. E. Armstrong publishes two articles on (l) 

 the conditions determinative of chemical change, and (2) the 

 nature of depolarisers. The former deals mainly with the 

 presence of water as a necessary condition of chemical change, 

 the latter with the question of the solution of metals in acids. 

 For some time past I have been engaged with work on the 

 former subject, upon terms of mutual understanding, with my 

 friend Mr. H. B. Baker, whose experiments, following upon 

 those of Prof. H. B. Dixon, have revolutionised our conceptions 

 of chemical change. In the last four years I have also carried 

 on investigati jns upon the reactions of metals with acids, es- 

 pecially nitric and sulphuric. I should, therefore, propose to 

 deal more fully in a separate publication with the interesting 

 speculations raised by Prof. Armstrong in the articles quoted 

 above. For it has become apparent ihat after a century of work 

 in chemical science we have no answer to the questions, (i) 

 What is the nature of chemical change ? and (2) What is the 

 cause of its commencement ? It is probable that both questions 

 resolve themselves, in the long run, into 'the first. Of facts 

 there is no end, but no interpretation thereof. 



The subject is, therefore, ripe for discussion, not only for 

 chemists among themselves, but also, as Prof. Armstrong aptly 

 remarks, for physicists. 



Such a discussion might be brought forward at the Chemical 

 Section of the British Association, at Nottingham, in the cur- 

 rent year, or, more appropriately, next year in Oxford, the home 

 of Robert Boyle, Mayow, and other earlier chemists. 



V. H. Veley. 



The University Museum, Oxford. 



