NA rURE 



381 



THURSDAY, FEBRUARY 22, 1894. 



BOLTZMANN ON MAXWELL. 

 Lectures on Maxwell's Theory of Electricity and Light. 

 Part ii. By Dr. Ludwig Boltzmann. 8vo. pp. 166. 

 (Leipzig: Barth, 1893.) 



THIS second part of Dr. Boltzmann's account of 

 Maxwell's electromagnetic theory is written from 

 a somewhat different point of view from the first part. 

 The first part presents the theory from the mathematical 

 point of view of a dynamical system whose generalised 

 coordinates are known. This one presents the theory 

 from the physical point of view of a continuous medium 

 whose intimate structure is indeed not fully understood, 

 but whose changes of structure can be fully represented 

 by certain vectors. Although Maxwell has presented 

 the subject from both points of view, the one which 

 really determines the form of his work, and that appears 

 to have led him in his investigations, is the physical view 

 of this second part. A purely analytical view is hardly 

 ever as suggestive as a physical and geometrical one. 

 This latter one suggests extensions, suggests advances 

 in a way that purely analytical investigations seldom do. 

 Compare, for instance. Ampere's investigation of the 

 action of elements of currents on one another with 

 Faraday's treatment of the same subject. The latter has 

 suggested the whole of the recent advances, the working 

 of the ether, the identity of light and electromagnetic 

 waves. The former was magnificent, brilliant no 

 doubt, but it was cold and dead. 



In the preface to this second part, Dr. Boltzmann ex- 

 plains the absence of diagrams, marginal notes, &c. 

 virhich appeared in the former part. He says a friend 

 conveyed to him the valuable criticism that "Your book 

 is dear." He has consequently left out the embellish- 

 ments that Englishmen love, as being too expensive for 

 the poor German student, and has only left the motto, 

 which costs nothing, to wit : — 



" War es ein Gott, der diese Zeichen schrieb 

 Die mit geheimniisvoll verborg'nem Trieb 

 Die Kriilte der Natur uiu mich enthiillfn 

 Und mir das Herz mit stiller Freude fiillen." 



And even this he has not taken from that classic lore 

 that Englishmen delight in. but from a German poet. 



Although he regrets the dark and inconsequent character 

 of much of Maxwell's work, he congratulates scientific 

 men that this has left them the more to do. For himself, 

 he only claims to be an exponent of Maxwell's views, and 

 hopes that he may succeed in helping students to under- 

 stand them. 



Electromagnetic equations lend themselves to a great 

 variety of interpretations by analogy with displacements 

 of a medium. They are a system of vectors related to 

 one another by a very simple method of derivation each 

 from the last, by the process of what Dr. Boltzmann 

 says the English call " curling." Starting with the vector 

 potential, the magnetic force is its curl, and the curl of 

 the magnetic vector is the time rate of variation of the 

 electric vector. Any one of this system of vectors may 

 be likened to the displacement or velocity of an incom- 

 pressible medium, and hence we have one system in 

 NO. 1269, VOL. 49] 



which the vector potential is so likened, one in which the 

 electric displacement is so likened, and one in which the 

 magnetic displacement is so likened. These latter two 

 analogies have been the favourite ones. Maxwell fre- 

 quently speaks in terms of the system in which the 

 electric vector is likened to a displacement of an incom- 

 pressible medium, and the likening of the magnetic vector 

 to a flow is quite common. In these systems electric energy 

 is generally considered as potential, and magnetic energy 

 as kinetic. Dr. Boltzmann, however, likens the vector 

 potential, which he calls the tonus, and which Mr. Oliver 

 Heaviside relegates to the realms of merely convenient 

 suppositions, to a displacement of the medium ; and in ac- 

 cordance with this analogy the magnetic energy becomes 

 potential, and the electric vector, which is proportional 

 to the rate of variation of the vector potential, being thus 

 a velocity, requires the assumption that electric energy is 

 kinetic. An obvious difficulty arises here from the 

 necessity of making an electric current an accelera- 

 tion which cannot of course be constant for ever. 

 In this connection it may be worth while observing 

 that the possible existence of one closed surface inside 

 another with static lines of the electric vector between 

 them makes it necessary to assume either (i) that the 

 vector potential represents a twist round its line, and not 

 a displacement along it ; or (2) that it is a displacement up 

 some lines and down others ; or (3) that there are sources 

 and sinks of the ether where there is electrification, be- 

 cause without sources and sinks we cannot have a con- 

 tinuous flow going on out from a closed inner surface to 

 a closed outer one. If the tonic vector be a twist, the 

 magnetic vector will be of the nature of a A% and it 

 would be this structure which should be elastically re- 

 sisted, and not a twist, as Dr. Boltzmann's and Mr. 

 Larmor's assumptions give. This would return some- 

 what to Mr. Glazebrook's proposal of years ago. Of 

 course, a complex change of structure, such as a com- 

 bination of I and 2, or any other change, such as crystal- 

 lisation in a hemihedral crystal, would be a possible solu- 

 tion. In a hemihedral crystalline form, because its two 

 ends must differ in sign. Gravity is probably due to a 

 change of structure produced by the presence of matter^ 

 which is analogous to a non-hemihedral crystallisation 

 because it is always attractive ; there seems no reason to 

 suppose that any bodies exist possessing negative gravita- 

 tion, the supposed levity of the old philosophers. 



To the vector potential, Boltzmann gives Faraday's 

 name of electrotonic state at the point, or, shortly, the 

 tonus of the element of volume. The rate of change ot 

 this tonus is the electric vector E, and the kinetic energy 

 due to it is the electric energy per unit vol. 



T = K/8 -K . E- 

 This tonic strain is accompanied in general by a tonic 

 stress depending on the curl of the tonic vector which is 

 the magnetic vector, H = curl E, and a corresponding 

 potential energy 



V= "- H-. 

 2 



All this is most interesting in connection with Mr. 



Larmor's recent papers. He uses Maxwell's analysis in 



which the magnetic energy is kinetic, and consequently 



assumes the magnetic vector to be a flow, which he has 



R 



