February 3, 1898] 



NATURE 



l^^l 



LETTERS TO THE EDITOR 

 \The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 vinnuscripts intended for this or any other part of NATURE. 

 No notice is taken of anonymous covimumcations.'l 



The Mathematics used in Connection with Physics. 



It may seem ungracious for an author to reply to a review 

 containing so many kind expressions as the one with which 

 Prof. Ayrton has honoured my book on "Electricity and 

 Magnetism " in your issue of November i8 ; nevertheless, I 

 trust that you will permit me to make a few explanations and 

 even corrections, if the word is permissible. 



Since the reviewer states that " it is not quite obvious what 

 is the object of giving" the mathematical introduction, I may 

 state rather more fully than I was able to do in my rather long 

 preface for what class the book was intended. The only class 

 of students with which I come in contact at Clark University is 

 composed of so-called "graduate students," that is men who 

 have taken the bachelor's degree at a college, and are intending 

 to undertake- research. It is not generally known that this 

 univeisity was founded for the express purpose of encouraging 

 such men, and consequently, alone among American universities 

 (with the exception of the Catholic University in Washington), 

 has no other students. The same class of students is, however, 

 to be found in large numbeis at all the larger universities, so 

 that what I have to say is of general application. These 

 students come to us from all parts of the United States and 

 Canada, and I have had two or three from Europe, so that 

 they have had very various training. They have been at 

 college for four years or more, and have generally taught for 

 awhile themselves. They may have studied the calculus for 

 two years or less, so that although they all know " the meaning 

 of a differential coefficient," and- are able to integrate and 

 differentiate with fluency, they are not Cambridge wranglers, 

 and their ideas regarding continuity, convergence of series, 

 definite integrals, and the like, are generally decidedly hazy, 

 while they probably have no acquaintance with the calculus of 

 variations, the theory of functions, and more difficult subjects. 

 It was in order to have some of these matters of most frequent 

 occurrence in a convenient place to refer to that I prefixed the 

 introduction, not with the idea of giving a complete treatment, 

 but to show the student some of the things he should certainly 

 get up, and by means of foot-notes to show him where he could 

 go further. In the numerous kind letters that I have received 

 from teachers of physics in this country, I think all have 

 especially commended the idea of this introduction. At any 

 rate, I had so good an example as Maxwell, who thought it 

 worth while to put a mathematical chapter at the beginning. 



With regard to the suggestion that " possibly the students of 

 Clark University, when listening to such lectures as are given 

 in this treatise, have the physical meanings of the various 

 mathematical processes explained to them," which the reviewer 

 intimates would be desirable, I will say that these students 

 have worked generally two years or more in a laboratory, 

 making the usual measurements, and that it is hardly necessary 

 to explain to them what a magnet is, or how a galvanometer 

 is constructed. In order to cover the ground, it was necessary 

 to condense, as the book was already larger than I intended. 



Prof. Ayrton asks, " Is it correct to say that 'following the 

 usage of the majority of writers, we shall denote ' Laplace's 

 operator (I forbear to write the signs of variation, which the 

 printer has put for round ^s) ' by A,' seeing that many writers, 

 including Thomson and Tait, use V" and Maxwell - V" ?" 



I was rather careful to find this out when writing the passage, 

 and to quote only from the works which I have at hand. I 

 may state that to the two authors named using v^ might have 

 been added the names of Lamb, Minchin, Routh, Basset, and 

 Rayleigh, while the notation A is used by Mascart and Joubert, 

 Duhem, Kirchhoff, F. Neumann, C. Neumann, Mathieu, 

 Boltzmann, Ilelmholtz, Clausius, Drude, Picard, Jordan, 

 Hertz, Klein, and by Poincare, who calls it la notation 

 habituelle. The notation A., is used by Lame, Somoff, 

 Boussinesq and Voigt, while Betti writes A-. I did not say 

 " the majority of English writers." 



That Ciauss's theorem " would be perfectly useless if gravita- 

 tional, electric, and magnetic forces did not vary as the inverse 

 square," I can hardly agree, seeing that it would be just as 

 true, and would have applications to the flow of heat, hydro- 



NO. 1475. VOL. 57] 



kinematics, geometry, and the theory of functions. It is just 

 as well for electricians to remember that the world was not 

 made for them alone. 



The reviewer is certainly labouring under a misapprehension 

 when he says " The statement of the general problem of electro- 

 statics, as given at the beginning of § 135, is insuflicient since, 

 as pointed out, any number of solutions could be given to it." 

 On the contrary it is explicitly pointed out, at the top of page 

 265, that there is but one solution, which is completely 

 determined. It is comforting, however, to learn that "the 

 method, however, which is indicated for the solution of the 

 problem is correct, and leads in a neat way to the conception of 

 coefficients of induction "; though the credit can hardly be taken 

 by the present writer, the method being taken from Betti's 

 well-known treatise given among the list of works made use of. 



With regard to d'Alembert's principle, I svill only say that 

 for the purpose for which it is introduced, namely the deduction 

 of the equations of motion of a system of particles, Hamilton's 

 principle, &c., it seems to me that it makes no difference 

 whether the internal forces appear or not. In any case I have 

 said more about the principle than Kirchhoff, who merely 

 writes down the formula, and Appell, who only says qui 

 signifie que, pour un d^placement virtuel arbitraire imprime 

 an point h r instant, la somme des travaux virtuels de la fotce 

 cCinertie et des forces reellement appliquees an point est nulle. 

 Practically the same statement is made by Thomson and Tait, 

 and repeated by Tait in his small book on dynamics. 



It seems to me that Green's function is hardly " divorced 

 from its physical application" when it is stated that Green 

 introduced it to solve certain problems in electrostatics. Its 

 physical meaning as the potential of a certain electrification is 

 also given. 



The reviewer states that the analogy between electrical and 

 magnetic phenomena is carried too far when the same letter is 

 used for specific inductive capacity on some pages and for 

 magnetic permeability on others. Also that "the beginner 

 might expect to find /x instead of which he finds e ; the meaning, 

 however, of this e does not seem to be given." As a matter of 

 fact, it is explicitly stated that n is used where it refers in- 

 differently to either the electric or magnetic quantity, while the 

 meaning of « is given in the next line to the one in which it 

 first occurs, on page 509. 



With regard to my " poking fun " at anything or anybody, I 

 beg leave to assure Prof. Ayrton that his statement that " the 

 fun is not intentional on the part of the author " is due to a 

 misa pprehension. 



Finally, to the suggestion that "a course of 'Lectures on 

 Mathematical Physics' may fitly contain explanations of the 

 physical interpretations of the equations developed without 

 running the risk of appearing to pander to the electrical 

 contractor," it may be replied that much depends upon the 

 point of view. To me the steam-engine or the dynamo are 

 interesting as examples in thermodynamics or induction. I am 

 well aware that this is not the usual view, nor do I suppose it 

 ever will be. I need not conceal the fact that none of my students 

 have ever become engineers. We have an excellent engineering 

 school in our city (which, by the way, is Worcester, and not 

 Webster, as your heading makes it), and we have no reason to. 

 try to duplicate the work done there. Some of its graduates 

 have come to us, and have done good work in physics or 

 mathematics, but they have dropped engineering. I say this, 

 not with the slightest wish to disparage engineering or engineers, 

 but to emphasise the fact that there are others to be considered 

 as well. A. G. Webster. 



Clark University, Worcester, Mass., December 15, 1897. 



A New Single Picture Pseudoscope. 



The principle of the stereoscope is so well known that it is 

 unnecessary to point out that two dissimilar pictures are required 

 of a special character in order to produce the stereoscopic or 

 solid effect. Consequently it may be imagined that to obtain a 

 stereoscopic effect with a single picture is an impossibility. 



It is clear that if the possibility exists a true stereoscopic 

 combination would not result, but one which would approximate 

 more or less closely to the truth. 



Many devices have been brought out in the hope of giving a 

 single picture a solid appearance, such for instance as a large 

 convex lens. All these devices, however, fail to give the desired 

 result. The illusion, so far as it goes, is simply a distortion of 

 the original picture. 



