ON THE THEORY OF INTEGRAL EQUATIONS. 409 



solutions of a large number of integral equations may be obtained 

 indirectly with the aid of the partial differential equation. Other 

 interesting integral equations are obtained by replacing k 2 by — k 2 , or 

 J by Y . Some of these may be solved by using the inversion formulae 

 for Fourier, Hankel, and Laplacian integrals. 



Equations of Volterra's type may also be obtained from definite 

 integral solutions of Laplace's equation ; thus Le Eoux l remarks that 

 if z(x,y,n) is a principal solution of the equation 



d 2 z dz . j'dz , 



,— . = a + b~ + cz 

 dxoy ox oy 



then 



I f(a)z(x,y,a)da 



will also satisfy the equation, and if this reduces to g(x) when y = y„ an 

 equation of Volterra is obtained for the determination of /(«). In the 

 case of the equation of Euler and Poisson 



d 2 z a dz _ b dz cz _ q 



dxdy x — y dx x — ydy (x — y) 2 



the particular solution of the type 



z = (x — y)"(x — a)'" 



can be used to obtain Abel's integral equation. This is really only 

 a modification of the method by which Poisson first obtained Abel's 

 equation. 



In many problems of vibrations a homogeneous integral equation of 

 the first kind is obtained when the boundary conditions are introduced 

 into a definite integral solution of a partial differential equation. Thus, 

 in the case of the equation of a vibrating membrane 2 



which is satisfied by 



ear oy 



: 



4 



e rtt.C0Sa + ! ,6in,>Y( r( ^ f( 



the condition that V = o on the boundary x = x{0), y=y("), gives the 

 homogeneous integral equation 



2" 

 — L«W»)oOB.+y<0Bin«]^ a ^ (| 



According to a method described by the author the values of k for which 



1 A males de fficole Nbrmale Su/p&rieure, 1S95, per. 3', t. 12. 



2 Other methods of treating the problem of a vibrating- membrane with the aid of 

 integral equations are given by Picard, Bend. Palermo, 1906 ; Boggio, Rend. Linnet 

 1907, p. 336; Bryon Hey wood. Thesis, Baris, 1908; Sanielevici, Annates de I'EeoL 

 Nor male Superiewre, 1909, pp. 19-91. 



