AND THEIK CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 437 



the square a < s ^ b, a ^ t ^ b is drawn, and the portion of the triangle s ^ t which 

 corresponds to each set of inequalities is marked with its number. 



(a,*) (U) 



(I) 



(m) 



(0,0,) 



(a,a) 



Fig. 3. 



By expressing /; (,s, () in terms of the functions and (f), it is easily seen that at each 

 point of the region (i) 



M -*<*) - 



that in (ii) h (.s, t) is everywhere zero, and that in (iii) 



In a similar way, or by a mere interchange of the variables ,s and t, the values of 

 h (s, t) in the corresponding divisions of the triangle s > t can be obtained. 

 Now, let $! (s), </>, (.s-) be continuous functions defined by 



(<t =S s < b); 



= (aj^s^t); 

 also let Q 2 (s), <f) 2 (s) be two others defined by 



6 2 (s) = (a < s S i), 



t/ 1 -S I (D ( tS* 1 / . 7 \ 



-- \ ' ' \ / I ft ^^ t ^c I \\ * 



"TT T r ~ ~, r~ \^^1 *^ " ^~* / ) 



i ( < s < 6) ; 



and, finally, let two functions & r (.s, ) (? = 1, 2) be defined in the square a S s ^ b, 

 a^t^b by 



