FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 175 



(24), (25) converges uniformly in a certain set of points belonging to the interval 

 (0, TT), so also does the other. 



11. Let us next suppose that the pair of boundary conditions for the interval 

 (0, TT) is B'. In this case it may be shown that 



(27) 



is limited for ^ rs TT, :< t S TT, and all positive integral values of n ; and that, 

 as n tends to oo, it converges uniformly in those parts of the square s TT, 

 < t IT, which correspond to \ts\^ij. It will be found that the normal functions 

 which satisfy the differential equation 



and the boundary conditions 



a = at s = 0, ^ = at s = n, (28) 



as 



are 



A/ - sin ^s, A/-sinf, ..., A/- sin (), ..., 



TT TT TT 



the corresponding values of /u, being 



The GKEEN'S function of 



- n />>\ 



- u (^/ 



for the boundary conditions (28), will be found to be F A (a, t). Hence 



r*(M)= S ^ l In (-i) *(-!)* 



= 1 ^ft ^-^ A. TT 



It follows from this and the result quoted in III., 19, that 



-X [K x (, - r A (, t)] *^ __ *. (*) *. () - sin (n- J) * sin (n- 



Since the results quoted in that paragraph may be shown to lead to 



lim -X [K* (*, -,r x (, )] = lim p oF A (.v, <) a (p, s, t) = 0, 



we prove from this, by the method employed in 9, that, for unequal values of s and t, 

 ('27) converges to the limit zero, as n tends to 03. 



