ON THE THEORY OF INTEGRAL EQUATIONS. 3S3 



CO CO CO 



[(x - z<) 2 + (y - </>) 2 + (* - ^) 2 ] 4 ~ 4» j J j 



— CO CO — CO 



j£-x) $ -x l )+(v-y) fo-y') +(? -*) (W) 



[(i-.-r) 2 + (>7-^p + (c- ^r) 2 ]- w-xy+( V - y r-+({-z i y]i mm 



CO CO CO 



[(* + a. 1 ) 9 +.(y -$ X Y + (* -r z 1 )*]* = - j f7ft J j 



— CO —CO 



( x + a) (x l + a) c?g d£ 



[(z+a) 2 + (v-y) 2 + (HffW+HT + M!, 



the second of which i3 due to Lord Kelvin. 

 It also follows from the equation 



te-g')* co ( f-aj'-HE-it')' 



* 



_ (J-x'l* c 



—CO 



that for positive values of t the kernel on the left is definite for values of 

 x and a; 1 in any real interval. 



Additional Besults. 



If a definite function g(s,t) is such that g(s,s) vanishes for one 

 value of s = s , then g(s ,t) and g(s,s ) are zero for all values of t and s 

 (Mercer). 



16. Multiplication Fortmolce and their Applications. 

 Borel has shown that if, 



CO CO 



f(x) = L-*'<t>(t)dt } g(x)=L-*>i(t)dt. . . (1) 



o 



then, with certain limitations, 1 



CO 



f{x)g{x) = ^e-*\{t)dt, 







Wh6re X(') = j*(« - r)Hr)dr. 



This result is very useful for solving equations of the type 

 a(/) = \ft{t)+ pUt - r)0(r)dr, 







for if we multiply by e~ xt and integrate between o and oo there is an 

 algebraic relation between the functions derived from «(i), Biti, k(t) by 

 formulas of type (1). 



1 The theorem has heen established under much less stringent conditions by 

 Broniwich. Infinite Series, pp. 280-283- 



