NA JURE 



241 



THURSDAY, JANUARY 11, 1906. 



THE EQUATIONS OF THE WAVE THEORY. 

 The Analytical Theory of Light. By J. Walker. 



Pp. xv + 416. (Cambridge: The University Press, 



1004.) Price 15s. net. 



MR. WALKER has written a valuable book, but 

 one difficult to review. As he says in his first 

 sentence : — 



" The Science of Physical Optics may be regarded 

 as comprising two fields of enquiry ; the one includes 

 the study of the physical properties of a stream of 

 light, the other comprehends the investigation of the 

 Mechanism by means of which the stream is pro- 

 pagated. These two divisions may be called re- 

 spectively the kinematics and the dynamics of the 

 subject. " 



It is with the first of these that Mr. Walker is 

 concerned ; a few experimental facts suffice to show- 

 that a stream of light may be represented by a 

 periodically varying vector, transverse to the beam, 

 and on this result, with an appeal where necessarj to 

 experimental facts, the treatment of tin- subject is 

 based. 



I he appeal to experiment is made as rarely as 

 possible, and as a result we have a book dealing with 

 .! physical subject which is almost entirely pure 

 mathematics. Such a book has its value, and in this 

 the yalue is a high one, for the author has discharged 

 the task he set himself in an admirable manner; but 

 owing to his severe restraint the book lacks interest 

 and ils difficulty is increased. It is not a text-book 

 of physical optics, but of the analytical theory of light ; 

 the light vector satisfies certain differential equations, 

 and the consequences of this are traced out with 

 -1 rare degree of completeness. It is a book to which 

 students who desire to know how far the mathe- 

 matical side of the wave theory has been carried, 

 .\h.ii are its limitations, and in what directions 

 advances are possible will usefully turn. This know- 

 ledge is necessary for the physicist who is more 

 interested in the dynamical theory, for, as Mr. 

 Walker points out, it forms the touchstone on which 

 optica] theories are tried, and no one theory of the 

 ether can at present be said to hold the field. No 

 doubt the introduction of even the salient points of 

 the various theories might have had the effect which 

 the author fears of veiling his main purpose; still, 

 1 lie restraint which he has laid on himself has its 

 disadvantages. 



Starting with the ordinary geometrical propositions 

 of the wave theory, we come in the second chapter 

 tn a discussion of Michelson's experiments and the 

 recent work on the nature of white light; the proper- 

 ties ul the polarisation vector are deduced from the 

 non-interference of two beams polarised in planes al 

 right angles. In connecting together the intensity oi 

 a beam of light and the amplitude of the polarisation 

 vector, a difficulty is at once met with unless we 

 know the relation between the energy of the stream 

 and the amplitude. For this purpose we require to 

 determine the nature of the yector, and this it is 

 NO. 1889, VOL. 73] 



impossible to do without advancing a theory of the 

 mechanism of the transmission, a course which is 

 closed to us. However, the assumption that the 

 square of the polarisation vector measures the in- 

 tensity is shown to lead to consistent results, and this 

 assumption is made. 



The analytical theory really begins in chapter v. 

 with the differential equations of the polarisation 

 yector ; the previous discussion has enabled us to 

 express this in the form of a function of vt — r, where 

 v is the velocity of propagation, and from these the 

 differential equations are deduced in the usual form, 

 and the important result that when the wave velocity 

 is independent of the period any singularity of phase 

 or amplitude travels with the speed of the wave is 

 shown to follow by a method of proof due to 

 Poincare; Lord Rayleigh's generalisation in the case 

 when the velocity is a function of the period follows. 



Reference had been made in an earlier chapter to 

 Huyghens's principle and its connection with the 

 rectilinear propagation of light; the full proof 

 of the principle depending on the relation at any 

 point within a surface S between a function <f> satis- 

 fying d 2 <j> /dt 2 = w 2 v 2 <j> and a certain surface in- 

 tegral taken over the surface of S is then discussed, 

 leading us to Stokes's well known law of the secon- 

 dary disturbance due to a wave of light, and also to 

 an expression for the disturbance in the neighbour- 

 hood of a screen producing diffraction. Fraunhofer's 

 diffraction phenomena are first discussed here; the 

 more complicated cases known as Fresnel's, in 

 which the screen is not at the focus of the light 

 forming the diffraction pattern, follow. The treat- 

 ment is based on that of Lommel, and deals fullv 

 with a rectangular aperture or obstacle, a straight 

 edge, Fresnel's biprism, and a circular aperture and 

 disc. The treatment of the biprism follows Struve 

 and Weber's work. 



In chapter ix. an account of some quite recent work 

 by Sommerfeld, Poincare\ and Macdonald dealing 

 with the application of spherical harmonics to diffrac- 

 tion phenomena is given, and after this a short 

 account of Newton's rings and of the laws of re- 

 flection and refraction leads to double refraction. For 

 uniaxial crystals the theory is based on Huyghens's 

 assumption, first satisfactorily verified by Stokes, that 

 the wave surface consists of a sphere and a spheroid 

 which touch at the extremities of the axis of the 

 latter, while for biaxial crystals all the laws are de- 

 duced from Fresnel's polarisation ellipsoid, a surface 

 which has the property that the velocities of pro- 

 pagation of any wave are equal to the reciprocals of 

 the axes of the section of the ellipsoid by a plane 

 parallel to the wave. Hence the form of the wave 

 surface and the laws of double refraction follow in 

 the usual way. 



Chapter xii. contains a number of results on the 

 wave surface which are not easily found elsewhere ; 

 in dealing with reflection at a crystal surface, the 

 equations of the polarisation vector in a crystal are first 

 formed, and then the surface conditions are deduced. 

 Free use is made of MacCullagh's ingenious device 

 of uniradial directions. The discussion of the inter- 



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