FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 1G5 



which leads to 



2 I ft (., ^ (.) rf . . 



4. Let D (X) be the detenninant of K (s, t), the GREEN'S function of 



for the boundary conditions B'. Then, in accordance with FREDHOLM'S theory, we 

 have 



llecalling that X = p a , we see from the equations (III., 6) that 



hence, applying the formulae of the preceding paragraph, 



Now let A (X) be the determinant of the GREEN'S function of 



<3?u 



ds 1 



+ KU = (K < 0), (10) 



for the boundary conditions 



^ = at s = 0, ^ = at s = IT (11) 



From the result obtained above we see that 



Thus we have 



If Xo = /a,,* is any negative number, we obtain from this, by integration,* the 

 formula 



. 



A (X) A 

 Since the integral 



tends to a finite limit as p tends to oo, we thus see that, as X tends to co, the 

 quotient D (X)/A (X) <i'H</s to a finite limit which is not zero. 



* The function a (p) is evidently integrable. 



