540 REPORT— 1902. 



them iuto undulations of convex, or of concave, or of flat wavelets. That a I'esolu- 

 tion into uniform fiat wavelets is always possible is proved in a paper published 

 at page 570 of last year's Report of the British Association,^ and the present paper 

 is devoted to explaining a convenient way of making practical use of this metl:od 

 of resolution. The resolution of light into undulations of uniform fiat wavelets 

 has the great advantage over every other resolution that each of these flat wavelet 

 components does not undergo alteration as it advances through the medium. 



Instead of maliing use of rectangular or polar coordinates, the position in 

 space and the motion of eacli of the component undulations are referred to a fixed 

 hemisphere, for which the name of reference hemhphere is suggested. Each point 

 on the couve.x side of this hemisphere is tbe guide-point of two undulations of flat 

 wavelets, one travelling outwards along the radius to that point and the other 

 travelling in the opposite direction. If, then, we know whether the undulation is 

 outward bound or inward bound, the position of its guide-point upon the surface 

 of the reference hemisphere indicates both the orientation of the wave-fronts and 

 the direction in which they are being propagated. 



These particulars are even more conveniently indicated by the orthogonal 

 projection of the guide-point upon the flat circular base of the hemisphere. This 

 projection may be called the index-point of the undulation, and the pattern pre- 

 sented by the index-points of all the undulations that we have occasion to deal with 

 in any problem may be called their indicator diagram. 



From this construction a number of propositions are deduced, which make 

 it easy to apply the above method of analysis to the solution of many optical 

 problems, including some which either cannot, or cannot without difficulty, be 

 treated by any of the usual methods. 



It also suggests interesting experimental verifications, and makes what is seen 

 in these experiments intelligible in a degree that they have not hitherto been. 



The same proof furnishes the more general tlieorem — that light traversing any 

 uniform transparent medium, whether isotropic or doubly refracting, may be 

 resolved into undulations of flat wavelets — and that the reference surface by which 

 the orientations, velocities, and states of polarisation of these undulations can be 

 indicated is, in general, half of the wave-surface of that medium. This reference 

 surface becomes a simple hemisphere, and at the same time loses its power of 



' At p. G17 of the Montldy Notices of the Royal Astronomical Society, issued in 

 September 1902, Mr. E. T. AVhittaker has given another and extremely elegant proof 

 of this theorem by showing tliat every solution of the equation 



can be analysed into undulations of flat wavelets advancing with the speed It. 



This proof has the advantage of deriving the resolution directly from the funda- 

 mental differential equation of wave-motion where the speed of propagation is the 

 same in all directions and is a constant as regards ,r, y, z, and t ; a condition which, 

 as Clerk Maxwell showed, is fulfilled bj^ electromagnetic waves in an isotropic 

 medium whenever we may assume that tlie product of the two inductive capacities 

 of the medium is independent of the intensity of the alternations of electromagnetic 

 stress, notwithstanding that in dispersing media it is not independent of their 

 periodic time. This is doubtless a correct assumption in the case of those electro- 

 magnetic waves whicli constitute any liglit tiiat our eyes can see. 



On the other hand, the proof given in the BA. Report of 1901, which is based 

 upon MacCullagli's Principle of Reversal, has two advantages of much importance 

 to physicists : that it furnishes useful details of the resolution, and that it exhibits 

 the relation in which the resolution into flat wavelets stands to neighbouring reso- 

 lutions into nearly flat wavelets, which are what practically have to be dealt witli 

 in making experiments. Moreover, the proof by the Principle of Reversal has 

 another considerable advantage in that it is applicable to doubly refracting media 

 in which 7^, the speed of propagation, is a function of the vector, as well as to iso- 

 tropic media, in which it is the same in all directions. It is fortunate, therefore, 

 that we are in possession of both proofs. 



