420 MR. J. MEitCEfc: FUNCTIONS OF POSITIVE AND NEGATIVE TYPE, 



two sets of four lines drawn parallel to the axes of s and t; these are the lines 

 t = s l 7 ) , * = a 1 (iy + e); = *ii7, a = i(i7 + ) respectively, and they may be 

 identified by observing that the number at the point where any one of them 

 intersects an axis is the value of the corresponding variable which is constant along 

 it. It will thus be seen that the square denoted by q n is bounded by the four lines 



S|_J|J 



c*x. 



H 



10 



AXIS OF S 



Fig. 1. 



s = SJ + TJ, f = Si + r) ; while the area d n , which is shaded in the figure, and which will 

 be referred to as the border of q n , is the part of the square bounded by s s 1 (? + ), 

 t Si (17 + e) exterior to q n . A. little reflection will show that, at points of Q which 

 do not belong either to q u or to d u , one or other of the functions d tiJI (s ; s^, B,^(t; s) 

 is zero ; that, at points of q u , each of these functions is unity ; and, finally, that in d u 

 neither function exceeds unity. It follows then that 



6. The integral 



= 1 in q u , 

 S 1 in d n , 

 = elsewhere. 



f K(s,t)e, i1l 



> a 



