ON THE THEORY OF INTEGRAL EQUATIONS. 367 



and has continuous second derivatives, then / is determined by the 

 formula 



/= - v 2 V. 



The Green's function was used primarily to solve the problem of 

 determining a potential function which takes given values V over the 

 boundary. The function G is now defined by the boundary condition 

 G = 0, and V is determined by means of the formula 



*rV(*,jr,«) = j^V(U0<«. 



A fundamental property of the Green's function is that it is a 

 symmetric function of the points (x, y, z); (£,/?,£) — i.e., 



G(x,y,z; ^,{)««G(Wi ®,yj). 



This means that the potential at P due to the charge induced over S by 

 a unit charge at Q is equal to the potential at Q due to the charge 

 induced over S by a unit charge at P. Helrnholtz's reciprocal theorem 

 in sound is also a consequence of this property of the Green's function. 

 The solution of v 2 V + &*V which satisfies the boundary condition is 

 given by solving the integral equation of the second kind. 



Y(x,y,z) = kj^G(x,y,z ; U0V(W)#*!#, 



s 



and the Green's function for the equation v 2 V + k 2 Y = is the solving 

 function of the non-homogeneous integral equation. The properties of 

 Green's functions in two dimensions are exactly analogous to those of 

 Green's function in one dimension, except that condition (1) is replaced 

 by the condition that 



G - log ^(x-Ey + (y-v) 2 



must be a regular analytic function in the neighbourhood of the point 



fc v). 



The Green's function has been determined for the circle, sphere, 

 circular disc, 1 spherical bowl, 2 cone, 3 box 4 and wedge. 5 



The idea of a one-dimensional Green's function was introduced by 

 Burkhardt in (1894), and has been developed by Hilbert and his 

 followers. 7 



The Green's function for a linear differential equation of order n is a 



'- 2 E. W. Hobson, Cambr. Phil. Trans. ; Stokes Commemoration Vol. (1899-1900), 

 p. 277. 



3 Mehler, see Heine's Thenrie der Kugelfunetionen, vol. ii., pp. 217-250. Also 

 H. M. Macdonald, Camb. Phil. Trans.; Stokes Commemoration Volume, p. 292.; 

 J. Dougall, Proo. Edinburgh Math. Soc. (1900). 



* A. Daunderer, Zeitschr. fiir Math. u. Pltys. (1909), p. 293. 



6 H. M. Macdonald, Proc. London Math. Soc, vol. xxvi., p. 161 (1895) ; A. 

 Sommerfeld, ibid. (1897). 



• Bull. Soc. Math., Bd. 22, 1894 ; See also papers by Dunkel, Bull. Amer. Math. 

 Soc, 1902 ; Bocher, ibid., 1901. 



' O'dtt. Nachr., 1904, and the dissertation by Mason (1903) ; Westfall (1905), 

 Anwils of Math., 2nd ser., vol. s. No. 4. An application to systems of differential 

 equation is made by Bounitzky. Liouville's Journal, t. 5, 1909, Fasc. 1, p. 65. 



B D 2 



