AND THEIR CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 423 



and that in the border d^ the last pair of equations still hold, but #,_, (s ; s a ), 6 ti1l (t ; s^) 

 are each less than, or equal to, unity. From this it appears that the function 



[>i0.., (* ; *i) + a- A, (*' : s *)] M., (t > *i) + *A, (* : *)] = We in '//. (> = !. 2 )' 



= outside the borders d^, 

 and that in the border d at3 its modulus is < | x^ 



dlf 



< 









u 



^f- 

 n 



8. Let us now write 



e(s) = x 1 #, <1l (8',s 1 ) + x a e, in (s;s a ), 



for the sake of brevity. It follows from the remarks of the preceding paragraph that 



( j K (,v, t) 6 (s) 9 (t) ds dt = 2 2 x^Xp I K (s, t) (ds dt) 



= 1 ~ ? afl 



+ 2 2 f K (s,t)0(s)e(t)(dsdt). . . (4) 



a=l 8=1 Jrf a/3 



Now the area of each of the borders d^ is 4e(2^ + e), and so we have 

 2 2 f K (s,t)e(s)0(t)(dsdt) 



o=l /J=l Id . 



