NA TURE 



49 



THURSDAY, NOVEMBER i8, 1897. 



THE MATH EM A TICS USED IN CONNECTION 

 WITH PHYSICS. 



The Theory of Electricity and Magnetism ; being Lectures 

 on Mathematical Physics. By A. G. Webster, Assistant 

 Professor of Physics, Clark University, Webster, 

 Massachusetts. Pp. xii + 571. (London : Macmillan 

 and Co., Ltd., 1897.) 



THE aim of the writer, according to the preface, in 

 the preparation of this volume has been to present 

 to the students the results of the Maxwellian theory as it 

 stands at present, after the labours of Faraday, Maxwell, 

 Helmholtz, Hertz, and Heaviside. 



Prof. Webster is a somewhat young man of marked 

 promise, whose contributions to the discussions in 

 Section A of the British Association formed one of the 

 features of the meeting of that section at Toronto this 

 year, and the book shows that he is fully capable of 

 appreciating the mathematical significance of physical 

 facts. It is doubtful, however, whether a student would 

 be able to appreciate the physical significance of the 

 mathematical theory from reading it. Possibly the 

 students of the Clark University, when listening to such 

 lectures as are given in this treatise, have the physical 

 meanings of the various mathematical processes ex- 

 plained to them. If not, they must possess an exceptional 

 amount of ability to enable them to arrive at correct 

 physical interpretations of the mathematical equations. 



The first fifty-two pages of the book are occupied with 

 a short summary of the principles of the vector calculus, 

 the definitions of variables, functions, differential coeffi- 

 cients, definite integrals, line and surface integrals, some- 

 thing about the calculus of variations, &c. It is not 

 quite obvious what is the object of giving this, since a 

 student who was capable of following the book would 

 understand the meaning of a differential coefficient 

 before opening it ; or if, on the other hand, he was 

 ignorant of the differential and integral calculus, he 

 would require far more detailed information than is 

 given in these fifty-two pages before he could tackle 

 triple integrals, vector differential operators, &c., which 

 are used quite early in the book. 



Is it correct to say that, "following the usage of the 

 majority of writers, we shall denote 



i: + ^ 





by A," seeing that many writers, including Thomson and 

 Tait, use v^ and Maxwell - V^ ? 



Gauss' theorem may undoubtedly be stated as a piece 

 of mathematical analysis, as it is done on pp. 75-78 ; 

 but the theorem would be perfectly useless if gravitational, 

 electric, and magnetic forces did not vary as the inverse 

 square of the distance. When, therefore, the theorem is 

 divorced from all physical application or interpretation, 

 is not the student less likely to grasp it and to 

 remember it ? 



The last section of the introductory chapter, which 



NO. 1464. VOL. 57] 



brings us to p. 91, discusses the theor>' of functions of a 

 complex variable with clearness and conciseness ; and 

 this section might with advantage have been e.xteftded, 

 and some of the earlier portion of the book omitted. 



Part i., which commences on p. 91, is on the " Theory 

 of Newtonian Forces"; but forces as realities, and not 

 merely letters which satisfy a number of equations, have 

 not much existence in the chapter. Take, for example, 

 d'Alembert's principle described on p. 107 This prin- 

 ciple is in reality the result of an argument based on 

 Newton's laws, and leads to a valuable set of equations 

 in dynamics. The reader, however, is not told this, but 

 only that the analytical statement of d'Alembert's prin- 

 ciple is given in Lagrange's equation of virtual velocities. 

 This equation, Mr. Webster rightly states, involves all 

 the internal forces, and so the student might not suspect 

 that the essence of d'Alembert's principle was the 

 elimination of these internal forces. 



The rest of the Part i. is good, the subject of attractions 

 being well and lucidly treated. Does not, however, the 

 introduction of the n axes of the harmonic of the nth 

 degree add unnecessary difficulty, and somewhat hide 

 the beauty of harmonic analyses ? For students who 

 already possess a good knowledge of magnetism, the 

 treatment of polarised distributions will be usefuL 



Although the title of this book, as given on the cover 

 is "Electricity and Magnetism," it is not until p. 243, or 

 nearly the middle of the book, that we come to the 

 portion that deals specially with electricity. The state- 

 ment of the general problem of electrostatics, as given at 

 the beginning of § 135, is insufficient since, as pointed 

 out, any number of solutions could be given to it. 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 from which the 

 coefficients of potential are deduced. 



§ 151, on Green's function, like much of the matter in 

 the book, is divorced from its physical application, so 

 that a student will hardly see the physical nature of the 

 problem that Green set himself to solve, or the method by 

 which he solved it. In fact, the author criticises Green's 

 work as follows : — 



"Reasoning depending on such physical facts was 

 frequently made use of by Green, and while not legiti- 

 mate for purposes of mathematical demonstration, is 

 frequently of service to the physicist." 



The discussion, however, of the application of conjugate 

 functions is good. 



It is not the custom in this country to call the conductor 

 of a Wheatstone's bridge, in which the galvanometer is 

 inserted, " the bridge wire," and we are not aware that 

 this practice is followed in America either. 



On page 357 the author defines /* as the inductivity of 

 the medium, and explains that this becomes specific in- 

 ductive capacity in the electric case, and permeability in 

 the magnetic. This analogy is, of course, quite correct, 

 but it is carried too far when the same letter \i is used for 

 specific inductive capacity on some pages and for mag- 

 netic permeability on others. In considering the di- 

 mensions of the electrostatic and electromagnetic systems 

 of units, the propagation of waves, &c., the beginner 



D 



