10 Mr Pollard, On the term by term integration 



exist and are finite and equal for X ^ a ; then both sides of (5) 

 exist and are equal. 



IT. If the double limit lim \ f{x, y)\dxdy exists 



X-*-ao , Y-^x J a J b 



and is finite; then without further condition both sides of (5) exist and 

 are finite and equal. 



Deductions from the general theorems. 

 9. A'. // f{x,y) = <^{x)d{x,y), 



where j 6 {x, y) dy \ < G for x^a, y ^b, 



Jh 



f{x, y) dy converges for x^a,' 



'b 



and \(f>{x)\dx converges; then both sides of (5) exist and are 



J a 



finite and equal. 

 B'. If either of 



1 dx \f{x, y) I dy, dy \f{x, y) \ dx, 



J a J b J b J a 



exist and are finite; then both sides of {5) exist and are finite and 

 equal*. 



rX rY rx r=° ■ 



C. If lim dxl f{x,y)dy, dx \ f{x,y)dy, 



Y-*-oo J a J b J a J b 



exist and are finite and equal, and 



dy fix, y) dx 



Jb Ja 



converges uniformly for a^ X, and 



o 



f{x, y) dx 



/ 



J a 



converges for y ^ b ; then both sides of (o) exist and are finite and 

 equal. 



* This is de la Valine Poussin's theorem. See Bromwieh, Infinite Series, p. 457. 

 The hypothesis given by Bromwieh to the effect that both the integrals 



/"OO /"OO 



are convergent is unnecessary, the existence of one (the one necessary to the 

 existence of the repeated integral) is sufficient. That of the other is implied by 

 the existence of the double hmit, see Note 2. 



