ON THE THEORY OF INTEGRAL EQUATIONS. 377 



The properties of the orthogonal system of functions associated with 

 an integral equation of the second kind with symmetrical kernel are 

 considered very fully by Schmidt. He shows how a complete set of 

 orthogonal functions for the integral equation may be constructed, and 

 obtains the important expansion 



rts) = f(s) +\2^{f( S ) % (s)ds 



a 



for the solution of.the equation 



6 

 a 



It is easy to derive from this expansion the conditions that a solution 

 may remain finite in the vicinity of a root X„. 



Schmidt has defined normal functions for an unsymmetrical kernel 

 k (s,t) by means of the equations 



0»(S) = U(s,t)Xn(t)dt, 



b 



X»(0 = Ajn(s)K(8,t)ds, 



and has obtained many theorems which are analogous to those proved 

 for symmetrical kernels ; for instance, the roots \ are all positive and 

 there is an expansion theorem. The functions <f>„(s), x„(i) are respectively 

 normal functions of the symmetric kernels of positive type, 



K(s,t)^(x,t)dt, K(x,sy(x,t)dx. 



There are a number of interesting cases in which the solving 

 function and determinant for one kernel may be derived from those of 

 another. Bryon Heywood and Goursat show that if 



K(s,t) = ^(s,*) + K. 2 (s,t), 

 where 



\< x (s,x)K 2 (x,t)dx = \i: 2 (s,x)i:i(x,t)dx =s 0, 



then 



D(\) = D.WD.W, 

 and 



K(M) = K 1 (s,0 +K 2 (s,t). 



This result is important in the discussion of the principal part of the 

 solving function in the vicinity of a multiple root. 



It has been shown by the author and by E. Schmidt, that when 



K (8,t) = h(s,t) + 2f n (s)g n (t), 



the solving function and determinant of k(s,t) may be calculated by means 



