422 RETORTS ON THE STATE OF SCIENCE. 



spectrum where use is made of a model atom in which electricity is 

 distributed over concentric spherical shells has been made by Jeans. 1 



The applications of integral equations to the theory of the spectrum 

 are at present in their infancy ; but there seems to be a very promising 

 field of research in this direction. It is possible for an integral 

 equation (or a set of linear equations with °°' unknown quantities) to 

 give an account of band spectra and continuous spectra. If matter is 

 discontinuous it is probable that Hilbert's theory of quadratic forms 

 and linear equations in an infinite number of variables should be of 

 primary importance. In this connection it may be mentioned that the 

 general expansion theorems for singular integral equations obtained by 

 Hilb and Wcyl seem to suggest a general dispersion formula of the 

 typo 



CO 



v°- = 6 + S-v 



1- 



which would be applicable when band spectra or continuous spectra 

 exist. In such a formula the function F(A) may be zero for a number 

 of intervals. 



The fact that a large number of types of integral equations or linear 

 equations in an infinite number of unknown quantities may be replaced 

 by a homogeneous integral equation of the form 



i 



[g(s,t)~\>.(s,l)] ,.(0(11=0 



may be of some interest in connection with the existence of different 

 series of lines in the spectrum of an element, for it is known 2 that in 

 many cases the values of A. for which an equation of this type can be 

 satisfied are the roots of a set of functions /*j(A), /< 2 (A.), ju; ( (A), and it is 

 possible that each of these functions may be associated with a series 

 of lines in the spectrum. 



The study of integral equations of this type promises results of 

 some interest. It may be worth while to investigate the changes which 

 occur in the values of the roots when a parameter 0, on which the 

 functions g{s,t) ^(s,t) depend, suffers a small change 10. The results 

 would probably have some application to the theory of the effect of an 

 increase of pressure or temperature on the lines in the spectrum. 



With regard to the other physical applications of Fredholrn's 

 equation, it may be mentioned that Poincaro 3 has treated some 

 problems in the diffraction of Hertzian waves, and W. H. Jackson 4 has 

 shown that a problem in the theory of radiation considered by Schuster 

 may be reduced to the solution of Fredholrn's equation. Applications 

 of Fredholrn's equation to the theory of the tides have been made by 



1 ' The Mechanism of Radiation' {Phil. Mag., ser. 6, vol. ii., p. 421). 



2 Mess. Math.. April 1910. 



3 Comptes rendus, 1900, t. 148, p. 449; Lesbwres at G'ottmgen, 1909; Teubner, 

 Leipzig, 1910. 



4 Bull. Amer. Math. Soc, 1910. 



