374 



RETT) TITS ON THE STATE OF SOTENOE. 



\[g{s,t)-\h(s,t)]+(t)dt = 



a 



with symmetrical kernels g(s,t), h(s,t). 



If the kernel is symmetric or belongs to the class just mentioned, it 

 can be shown that there is at least one root of the determinant 

 D(A) = 0, and consequently at least one solution of the homogeneous 

 equations. Also if the equation 



L(s,t)l3(t)dt = 0, 



\a(s)K(s,t)ds 



= 0, 



only possess a finite number of linearly independent solutions, there are 

 generally an infinite number of roots of the determinant D(A) = 0. 



If the kernel ic(s,t) is a symmetric function of s and t the above 

 theorems can be proved in a simple way by considering the energy 

 function of the integral equation 



f(s) = f(s)-\^(s,t) f (t)dt. 



a 



This function ?r(A) is defined by the equation 



«(A) 



f(s)f(s)ds 



and is such that 



d 



Xw(\) 



} [*)] 



ch. 



It follows from this equation tbat \ir(\) increases continually with X. 

 It generally becomes infinite at a value X„ which is a root of the 

 determinant D(\), an exception arising when the function <j>(s) does not 

 become infinite there. It should be noticed that although <f>(s) is 

 indeterminate as far as the addition of an arbitrary multiple of the 

 function </>„(.s) corresponding to the root A„ is concerned, yet ir(\) lias a 

 unique value on account of the equations 



b 



[f(s),j >u (s)ds=0, 



which express the conditions that f(s) is finite. 

 may be used for the following purposes : — 

 1° To find limits for 



K(s,t)a[s)u(t)dsdt 



\[a(s)]*ds 



1 

 ~'K 



The energy function 



