ON THE THEORY OP INTEGRAL EQUATIONS. 407 



differential equation is employed. 1 An integral equation is obtained as 

 soon as the boundary conditions are introduced. 



In some cases we have a single equation which has to serve for the 

 determination of two unknown functions. Equations of this type were 

 first considered by Cauchy in his memoir on the theory of waves, 1815, 

 An interesting example is mentioned by Forsyth in his presidential 

 address to the London Mathematical Society. 2 In many cases an 

 equation of this type may be reduced to the ordinary form or to a system 

 of equations by a special artifice. Systems of integral equations of 

 various types are also considered by Murphy. 3 



A few examples of Fourier's method may be mentioned here. Lord 

 Kelvin * used Laplace's solution, 



CO 



d 2 V 8V 

 of the equation of the conduction of heat, -,- 2 = ^rr, to solve the 



problem of the conduction of heat in an infinite solid with a plane face. 

 He assumed f(y) = for y <0, V = g(t) when x = 0. He was thus led 

 to the equation 



g(t) ^ f f@z</t)e-**de 



for the determination of /. This was solved with the aid of Fourier's 

 theorem, but it may also be reduced to an equation of Laplace's type. 



Schliini 5 afterwards obtained the same equation, and expressed the 

 solution in the form 



** - *| r-'(-£). fc 



— CO 



Lord Kelvin's solution is 



CO CO 



f(x) = A dm cosh x-/m cos x</m cos 2mt g(t)dt. 



o o 



The equation 



dz 2 dp 2 p dp p 2 



1 This method may perhaps be justly ascribed to Fourier, as it is used by him in 

 his ' Analytical Theory of Heat.' Poisson gave the solution of many partial differ- 

 ential equations in the form of definite integrals, but endeavoured to avoid the 

 c ccurrer.ee of integral equations. His efforts met with brilliant success as he obtained 

 definite integral solutions which are exceedingly well adapted for the initial conditions 

 occurring in the problems. 



2 Proc. London Math. Sec., ser. 2, vol. iv., p. 431. 



3 Camhr. Phil. Trans., vol. v. (1834). 



4 Cambr. Math. Journ. (1842-43), p. 170; Math, ami Phys. Papers, i , p.,10. 



5 Crelk's Journal, Bd, 71-72 (1870). 



