Resolution of Light into Undulations of Flat Wavelets. 265 



in other words, the solution is brought into a form which by 

 Fourier's theorem can be expanded into terms each of which 

 represents an undulation of fiat wavelets and is also a 

 particular solution of equation (1). 



2. Mr. Whittaker's proof has the advantage of deriving 

 the resolution directly from the fundamental differential 

 equation (eqn. (1)) of those wave-motions in which the speed 

 of propagation is constant as regards a, y, z and t — a con- 

 dition which, as Clerk Maxwell showed, is fulfilled by electro- 

 magnetic waves in an isotropic medium whenever we may 

 assume that the product of the two inductive capacities of the 

 medium is independent of the intensity of the alternations 

 of electromagnetic stress, although in dispersing media 

 it is not independent of their periodic timeo (See Clerk 

 Maxwell's ' Electricity and Magnetism,' § 786.) This con- 

 dition is doubtless complied with by those electromagnetic 

 waves that constitute any light that our eyes can see, 

 if we may omit from consideration the absorption of part 

 of the light by the medium, as we legitimately may when 

 dealing with journeys of any length in the open setber, or with 

 journeys of moderate length across transparent media. 



3. On the other hand, the proof by the Principle of 

 Reversal has an advantage which is of great value to the 

 physicist, namely, that it not only proves, like the analytical 

 method, that a resolution into flat wavelets exists, but further, 

 exhibits in each individual case details of the resolution and 

 of the relations in which this resolution stands to other neigh- 

 bouring resolutions : amongst which the most useful are its 

 relations to resolutions into concave or convex wavelets that 

 are nearly flat. It thus furnishes the general solution of 

 eqn. (1) which is presented in eqn. (2) : along with details 

 that are of practical value ; and further, with what may be 

 regarded as equivalent to a solution of the theorem in the 

 Calculus of Variations which would investigate the relation to 

 one another of the various functional forms which the solu- 

 tion of eqn. (1) can assume : of which functional forms 

 eqn. (2) is one. This great additional insight, and the adapt- 

 ability of the method to individual instances, offer such 

 assistance to the experimental physicist that it is fortunate 

 that we are in possession of both proofs. We shall be in a 

 better position to appreciate the advantages here spoken of 

 when we come to the experimental verifications. An account 

 of some of these is in preparation, and the author hopes to 

 publish it as a sequel to the present paper. 



4. The resolution of a given distribution of light into its 

 component undulations of flat wavelets is unique in the same 



