350 REPORTS ON THE STATE OF SCIENCE. 



Kirchhoff 1 made an interesting application of Fourier's formulae 

 when establishing the law of radiation which bears his name. In order 

 to show that the law is valid for each separate wave-length he had to 

 prove that the equation 



CO 



ism* P cj,(k)d\ = o 







could not be satisfied for every positive value p of the thickness of a 

 transparent plate unless </>(A) was identically zero. The proof depended 

 on a use of Fourier's formula. 



Fourier's formulae have many interesting applications in physical 

 optics. 2 Some of these depend upon the use of a formula due to 

 Eayleigh 3 and extended by Schuster. 1 



If 



CO C : 



k x (x) — cos xtf(t)dt Bj(j;) = sin xt<p(t)dt 



— CO — CO . 



CO CO 



Ao(.r) = cos xt^(t)tU B 2 (x) = sin xtifr(t)dt 



then 



I 



nm)dt= 1 



TV 



[A,(js)A,(aj)+B,(a;)B,(aj)]d«. 



Schuster has made an extensive use of this formula in a study of 

 interference phenomena. The formula has been extended to integrals 

 analogous to those of Fourier by H. Weyl/' 



Some very interesting applications of Fourier's theorem to the 

 evaluation of definite integrals containing Bessel's functions have been 

 made by Macdonald. 



Extensions of Fourier's Formula;. 



The following generalisation of Fourier's theorem was obtained by 

 Liouville 7 and Hamilton. 8 If 



X 



\p(x) = </j (x) i l.r 



is such that ijs(x) never exceeds a certain value and 



1 Ann. Phys. Cheni., 1860, 109, p. 275. 



* See, for instance, E. T, Whittaker, Monthly Notices of the Royal Astronomical 

 Society, Nov. (1906), p. 85. Sir J. J. Thomson, Phil. Mag-, vol. xiv. (1907), p. 217. 

 A. Eagle, Phil. Mag. (6), vol. xviii. (1909), p. 787. These three papers deal with the 

 radiation from hot bodies. 



3 Phil. Mag., 1889. 4 Ibid., 1891. 



5 Dissertation G'othnqen, 1908. 



6 Proc. Land. Math. Soc, vol. xxxv. (1903). 



7 Liouville's Journal, 1836, vol. i., pp. 102-105. s Irish Trans., 1813, vol. xix. 



