February 15, 1894] 



NA TURE 



359 



one P along the axis, and the other R at right angles to 

 the axis, in a plane through the axis and the point 

 considered, while the magnetic force H, say, has a 

 single component perpendicular to the plane. Thus two 

 differential equations are got connecting P, R, and H, 

 from which (P having first been found from the differential 

 equation involving P alone) R and H are found at once 

 in the forms MaP/ar, NaP/ar where M and N are multi- 

 pliers, and ;- is the radius drawn from the axis to the 

 point considered. 



It is to be noted also that the sign between the two 

 groups of terms into which Kj,(.v) is divided in (2), p. 263, 

 should be the same as that before log x in the first group 

 in brackets and that C should be taken with the same 

 sign as log .t-, and log 2 with the opposite sign. This 

 involves a correction likewise in the table of approxi- 

 mate values of the functions given lower down on the 

 same page. Again, the same constant C, which has the 

 value 



Lt (t^~ -\ogn 



is called Gauss's constant at p. 263, while the quantity 

 •y = ^"^ is called Euler's constant at p. 430. The estab- 

 lished usage seems to be to call C Euler's Constant from 

 its discoverer, who gave its value (to sixteen places of 

 decimals) in his histitutiones Calculi Differentialis. 



The " throttling " of the current in wires subjected to 

 rapidly alternating electromotive forces is fully considered 

 for a cable with inner and outer coaxial conductors, and 

 for two flat strips in parallel planes with a stratum of 

 insulating material between them. In this connection 

 the author first introduces Mr. Oliver Heaviside's word 

 impedance. Writing E for the external electromotive 

 force, I for the total current, and R and L for the effective 

 resistance and self-inductance, we have (p. 272) 



E = L^i + RI. 



a/ 



R is called by Prof Thomson the impedance. Accord- 

 ing to Heaviside's proposal it is .^R- + «^L- that should 

 be called the impedance, where n = 27r/T, T being the 

 period of the alternation. 



The manner in which the damping out of the vibration 

 is taken account of by the complex analysis is well 

 worth remarking. The eating up of the energy and con- 

 sequent tapering off of the amplitude according to an 

 exponential function of the distance from the starting end 

 by the impinging of the oscillations in the dielectric on 

 the conductors bounding it, and the lowering of speed of 

 propagation of phase in the dielectric below the natural 

 speed, that of light, all come out in the most beautiful 

 manner. 



Mr. Heaviside's careful synthetical explanations of 

 such phenomena are well worth reading in this- con- 

 nection. 



The author next passes to his own most valuable 

 investigations regarding the effect of subdivision of iron 

 j on the dissipation of energy in the iron of a transformer, 

 I to electrical oscillations on cylinders and on spheres, and 

 1 other problems of the greatest interest to all students of 

 ' the later developments of Maxwell's great theory carried 

 out by Hertz, now, alas, to be continued entirely by other 

 hands. 



NO. 1268, VOL. 49] 



The concluding portion of the book consists of a most 

 valuable account of the work of Hertz, and forms the 

 most appropriate supplement to Maxwell's great work 

 that could have been written. The idea of Faraday 

 tubes is well applied to picture the action of a Hertzian 

 resonator in its different positions relatively to the 

 vibrator in the experiments on direct radiation, and 

 those on waves along wires. Not only is Hertz's own 

 work fully described and explained, but the vast amount 

 of fine work that has been done at Dublin, Liverpool, 

 at Cambridge, and on the continent, is discussed, and 

 much of it submitted to careful mathematical analysis. 



Space does not permit of even a summary of the topics 

 here treated, and we can only say that the reader who 

 wishes to know these things well, and who shrinks from the 

 labour of digging them out of Proceedings, Annalen, and 

 Berichte, here, there, and everywhere, ought to read Prof. 

 Thomson's work. Such a work is worthy not only of 

 the author, but of the researches of the master and his 

 great disciple who have passed away. A. Gray. 



GREENHILUS ELLIPTIC FUNCTIONS. 

 The Applications of Elliptic Ftmctions. By Alfred 

 George Greenhill, F.R.S., Professor of Mathematics 

 in the Artillery College, Woolwich. (London : Mac- 

 millan and Co., 1892.) 



IT would be difficult to exaggerate the part which the 

 study of elliptic functions has played in the pure 

 mathematics of the present century. And this was to be 

 expected ; for whether we regard natural science as the 

 application of common sense to the material needs of 

 life, or as the outcome of the need for expansion in the 

 mental world, and whether we consider mathematics as 

 that exact basis without which progress was not per- 

 manently possible, or esteem it to be those higher Alps — 



Where we can ever climb, and ever 

 To a finer air — 



in either case we must see that a development of integral 

 calculus — a development which was competent to fill so 

 large a part of Legendre's life, which suggested such 

 magnificent algebra as we find in Jacobi's Fundamenta, 

 which promised, too, in Abel's hands such generalisations 

 as are not even yet brought to perfection, such a theory, 

 surely, was well worthy of persevering pursuit. And if 

 we attribute the present extent of the theory of curves 

 and of the theory of functions to the day when Rieniann 

 stood best man to the ideas of Cauchy and the sugges- 

 tions of hydrodynamics, we must admit it was because his 

 methods were employed upon the materials left by Abel 

 that such results have come. 



The importance of the present work lies in its recog- 

 nition that the theory of elliptic functions arose as a deve- 

 lopment of integral calculus, and as such may be expected 

 to supply a formulation of the solution of many problems 

 of physics otherwise regarded as unfinished. Prof. 

 Greenhill is well known to be aman who hasnot allowed 

 his unwearied application to such problems to destroy 

 his sympathy with pure mathematical speculation ; on 

 the contrary, he has sought, by every means in his 

 power, to fill the difficult position of apostle to the 

 Gentiles in this respect, by making as many of the results 



