DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
357 
Character of the /Ether Motions in nearly Homogeneous Light. 
§ 59. It has been thought that the possibility of producing a large number of 
interference fringes from white light is an indication of a certain regularity in the 
tether motion corresponding to such light. This view has been abundantly refuted 
by Gouy, Rayleigh, and Schuster. # The fringes cannot be produced without the 
use of a spectral apparatus ; and the number of the fringes is an index, not of the 
regularity of the white light, but of the resolving power of the spectroscope. 
A large number of fringes can also be produced without a spectroscope, by using 
radiations which naturally possess a high degree of homogeneity. The number of 
these fringes is a test of the homogeneity ; and in this case, it is also a test of the 
regularity of the tether motion. 
§ 60. We have justified the representation of light by a Fourier integral of the 
form 
co 
R cos ( ut + xp)clu, 
Jo 
where R, i p are functions of u. 
It can be shown that, for light of long duration, \Jj will fluctuate rapidly in terms 
of u. 
For approximately homogeneous light of mean frequency p, we will use the 
notation 
j R cos (p + ut + \p)du. 
In this expression R will be insignificant, except for values of u small compared 
with p. 
The above integral can be written 
cospt JR cos (ut + \\i)du — sin pt\ R sin (ut + f)du. 
The ranges of integration will practically be confined to a region on either side of 
zero, small compared with p. 
Each of the integrals 
J R cos ( ut + xfj)du, | R sin (ut + \fj)du 
is a function of t, whose variations are slow compared with those of cos pt. The 
expression cos pt JR cos (ut -fi xp) du may be taken to denote a simple vibration of 
period 2-rr/p, whose amplitude varies slowly. The zeros of the expression will be, 
effectively, the zeros of cos pt. Similar statements will apply to 
sin pt \Pt sin (ut + i p)du. 
* Gouy, ‘ J. de Ph.,’ series 2, vol. 5, p. 354; Rayleigh, Art. “ Wave Theory,” ‘ Encycl. Brit.’; ‘Phil. 
Mag.,’ vol. 27, p. 460, 1889; Schuster, ‘ Phil. Mag., vd. 64, p. 509, 1894 
