390 REPORTS ON THE STATE OF SCIENCE. 



In some eases the equation (1) possesses a unique solution ty{t) of a 

 specified type. If f(t) is continuous, this is the case for instance if the 

 double integral 



lb 



if 



a a 



;(x,t)<l>(x)(j>(t)dxdt 



is positive for every continuous function <]>. A kernel x(x,t) which is 

 symmetrical and possesses this property is said to be definite (ililbert). 

 The same definition appears to hold if 



b b 



L(t)dt and kHt)] 2 dt 



exist. 



When the solution of an integral equation can be expressed by 

 means of a single formula, e.g., 



<j>(t) = \v(t,x)f(x)dx 



this formula is called an inversion formula. A large number of inver- 

 sion formulae for different types of integrals are known. 



The integral equation (1) and the homogeneous equation (2) may be 

 studied as limiting cases of the integral equations of the second kind— 



b 



f(x) = nf(x) + Ux,t)<p(t)dt 



fi<b(x) + Ux,t)<l>{t)dt = 



when /< = 0. This method was suggested by Fredholm and has been 

 developed by the author. 1 



Many properties of the integral equation are suggested when it is 

 considered as the limit of a system of linear algebraic equations. There 

 are indications of this idea in the work of Fourier and Murphy. The 

 method was used by Volterra in 1884 and has formed the basis of the 

 epoch-making work of Fredholm and Ililbert in the theory of the 

 integral equation of the second kind. At present the most powerful 

 method of solving integral equations of the first kind is to express the 

 kernel as a definite integral of a convenient type and solve the equation 

 in two steps. The formulas of Fourier, Hankel, and Pincherle and the 

 multiplication theorem of section 16 are of fundamental importance in 

 this connection. 



15. Canonical Form of an Integral Equation of the First Kind. 



Definite Functions. 

 An integral equation of the type 



f(x) =L(x,t) f (t)dt 



a 



1 Mm. Matlu, April (1910), 



