2 Mr Pollard, On the term by term integration 



Conditions for passage to the limit under the sign of 

 integration, the range of integration being finite. 



2. We give, for the sake of reference, the two principal 

 elementary conditions. 



(C 1) If u^ (sc) is positive for a^x ^b; n = l, 2, 3 . . ., then if 

 either side of (2) is finite the equation holds, and if either side is 

 infinite both are. 



V 



(C 2) Ifl'^Un (oc) I < i|r (a;) for a-^a:^b,v=l,2,S,..., tuhere 

 n=l 



'>^ is summable in (a, b), then both sides of (2) eccist and are finite 

 and equal*. 



Resume of theorems of double limits. 



3. As the use of double limits is fundamental in the theory 

 about to be developed, we give a short summary of the results 

 required. 



(a) If the double limit lim S^, y exists, and lim 8x, y exists 



for all sufiiciently large y ; then lim (lim S^, y) exists and is equal 



to the double limit. Similarly for the limit lim (lim S^^y). 



(^) If Sx, y is increasing in x and y, and any one of 

 lim Sx,y, lim (lim *S^a;, j/X lim (lim>Sa;, j,) 



exist; then all three exist and are equal. 



(y) If 8x, y can be expressed as the difference of two functions 

 S'x, y, S"x, y ^^ch of which is increasing in x and y and 



lim {S'xy + S"xy) 

 X-»-oo , y^"Xi 



exists and is finite ; then 



lim Sx,y, lim (lim Sx,y\ lim (lim Sa;,y) 



all exist and are finite and equal. 



The condition (7) is especially convenient when 



S.,y=n''fil,v)d^dv. 



J aJb 



* De la Valine Poussin, Cours d^analyse infinites imale, t. i. , 3rd Ed., p. 264, 

 theorems iii and 11. 



