ON THE THEORY OP INTEGRAL EQUATIONS. 421 



(a - \ 2 P )u(x,y,z) = *(£,!/,£; x,y,z)u(£,ri,!;)cUchid; 



and since $ is a symmetric function it can be satisfied for the real 

 values of \ which are given by the transcendental equation 



The values of X have a point of condensation at V^/p. The special 

 assumption (3) is probably not necessary, as the values of X for which 

 (2) can be satisfied will also obey a law of the type required. The 

 properties of the more general equation (2), however, have not been 

 thoroughly investigated. 



Fredholm's theory of the spectrum has been developed by Schaofer, 1 

 who obtains a dispersion formula by introducing a forced vibration with 

 an ^-component of the type 



X(E,n,Z,t) =5 U(fo,0 cos fit. 



This leads to a non-homogeneous integral equation of the type 



p.Tf(x,y,g) = u(x,y,s) - Jfj*(£»v,f J v,y,z) u(£,r,,Z)dfcnd{; 



where 



1 



and Schmidt's expansion theorem gives 

 U{x,y,z) =p . J](x,y,z) + H p a ^^[[[w^)U^^)dUndK 



where 



1 T« 2 



/*« = _ .. 2 



T„ being a fundamental period of vibration and p the density. By 

 introducing an assumption of the form 



D x . = cos nt\T3($,y,z) + a\ \\u(Z,ri,Z)dt,dqd,] 



for the ^-component of the dielectric displacement, and supposing that 



XJ(x,y,z) does not vary appreciably over* the region of integration, 



Schaefer is finally led by means of Maxwell's equations to a dispersion 



formula of the type 



co M« 



p* = b + SeT - T - 



i 1 — ^ 

 T a 



where v is the refractive index, T. a natural period of vibration, and T 

 the period of the external force. 



Another application of an integral equation to the theory of the 



1 Ann. d. Phytii, 1900, Bd. 28, p. 121 ; 1000, Bd. 20, p. 715. Schaefer simplifies 

 the work by considering an analogous problem in one dimension. His analysis i» 

 applied here to the three-dimensional problem, 



