Flat' Wavelet Resolution. 595 



Xo material of which a lens can be made is perfectly trans- 

 parent, and accordingly the first thing to be done is to con- 

 ceive absolutely transparent glass substituted for the actual 

 glass ot the lens. It is the luminous disturbance which 

 would occur within this transparent medium that we propose 

 to resolve. We may, moreover, confine our attention to light 

 ot" one wave-length \, since any other wave-lengths that are 

 present can be dealt with in the same way. 



48. Let us next picture a superficial stratum of the lens 

 which need not be more than one or two wave-lengths in 

 thickness. Within this layer there will be " turmoil," i. e. 

 motion differing from ordinary wave motion, due to the light 

 passing from one medium into another, of the kind which 

 Stokes investigated in certain cases (see Stokes's Collected 

 Papers, vol. ii. p. 56). It' this surface layer be divided into 

 minute elements P 1? P 2 , &c, then it is legitimate (as in 

 Huygens' construction), and it will be convenient, to regard 

 the light within the lens as consisting of undulations of 

 spherical waves emanating from these P's. By this means 

 we include with the light coming from abroad, and without 

 making the investigation more complex, the special surface 

 action which takes place when light passes from one medium 

 into another. The disturbance we shall resolve will then be 

 the actual luminous disturbance which would occur in the 

 inside of the lens, excluding its surface layer, if the glass were 

 perfectly transparent. 



•49. If. as will usually happen, the light from each punctum 

 of the objects that transmit light to the lens, is not uniform, 

 but suffers from one minute portion of time to another, abrupt 

 or continuous changes, then the same will be true of the 

 " motion " in each of the P's. But this need give no trouble. 

 The light from a given punctum of the object which reaches 

 one of the P's — suppose P x — may in the first place be resolved 

 into light polarized in two planes at right angles to one 

 another ; and irregularities in the periodicity of the vibrations 

 in each of these directions are susceptible of being analysed 

 by that extension of Fourier's theorem which admits terms 

 with periods that are incommensurable. This analysis will 

 furnish a series of terms each of which represents regular 

 periodic vibration. Accordingly in our further resolution it 

 will be sufficient to deal with that one of these which produces 

 waves in glass of wave-length \. To simplify matters, we 

 may suppose all the P's to be annihilated except that one 

 whose emissions we arc going to analyse. Furthermore, we 

 shall make the hypothesis that the transparent glass of which 



