DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
331 
Page 
§§ 65-68. Examination of his argument . ..360 
§ 69. In a frictionless vibrator approximately monochromatic light will arouse 
natural periods of vibration in continually-increasing intensity . . . . .361 
§§ 70-71. Solution of the problem by means of Fourier’s integral.361 
Introductory. 
§ 1. The object of the following work is to make some progress with the mathe¬ 
matical representation of the motions which go to compose natural light. 
§ 2. It has always been recognised that interference phenomena forbid us to regard 
any natural radiation as consisting of an unending train of simple waves, such as may 
be represented by sine functions. At the same time, the equations of optics find 
their simplest solution in circular functions. It is desirable to enquire how far we 
may resolve a natural luminous motion with a sum of simple wave-trains by means 
of Fourier’s “Theorem of Double Integrals.” This procedure was first suggested by 
GouyA 
§ 3. Doubts have often been entertained as to the permissibility of this process. 
Writers have been sceptical as to the physical meaning and independence of the simple 
waves thus introduced. In the following pages will be found an attempt at a strict 
justification of the method. It is based upon two principles (i.) that we are cognisant 
of light only by means of the integral effects produced by the light during an 
interval of time w T hich depends upon the nature of the detector in use (the eye, a 
photographic plate, &c.) ; (ii.) that we are not concerned with simple wave-lengths, 
but rather with short ranges of wave-length, whose integrated energy we observe. 
The former principle is generally accepted ; the latter has been put forward with great 
force by Lord Rayleigh, f 
§ 4. In what follows we shall deal solely with plane and plane-polarised light. 
§ 5. The matter at issue cannot be introduced better than by a quotation from 
GouyJ :— 
“ On sait que la theorie ondulatoire, dans les explications qu’elle donne des 
phenomenes optiques, a pour objet immediat le mouvement simple , dans lequel la 
vitesse vibratoire|| d'un point quelconque est donnee par une equation de la forme 
v = 
* Gouy, ‘ Journ. de Physique,’ ser. 2, vol. 5, p. 354. (1886.) 
t Lord Rayleigh, ‘ Phil. Mag.,’ vol. 27, 1889. 
f Gouy, ‘ J. de Ph.,’ ser. 2, vol. 5, p. 354. 
|| It is clearly immaterial whether we speak of velocities and displacements of an elastic medium, or of 
electric and magnetic forces. The real objects of discussion are vectors, which can be interpreted in 
various ways, 
2 u 2 
