256 Proceedings of the Royal Society of Edinburgh. [Sess. 
can find a positive continuous function U(x , y) such that J“U(x , y)dx is 
a continuous function of y, and also 
|u(x,y)| <U(x,y) 
for all values o/ x > a and each value of y, then u(x , y )dx is a con- 
tinuous function of y. 
We have stated and shall prove this theorem for a single variable y; 
the argument is, however, quite general, and applies equally when there 
are n variables y. We may add, it also applies when there are m variables 
x, the integral being then the m-ple integral, and proper limitations are 
made with respect to the nature of the infinite region of integration. 
The integral in question is, in fact, the limit of the monotone ascending 
sequence of continuous functions fi,f 2 , • • • • where 
fn(y) = \l n u(x , y)dx - \l n U (x , y)dx , 
a , x 1 , x 2 , . . . . being a previously selected monotone ascending sequence 
of quantities having +oo as limit. 
The integral is also the limit of the monotone descending sequence of 
continuous functions g 1 ,g 2 , • • • • where 
9n(y) = }*”“(» , y)te + U(* , y)dx . 
Hence the result follows. 
§ 10. The theorem of § 5 gives us the following obvious modification of 
the tests of the preceding article for continuity at a point. 
Theorem . — If 
U l + U 2+ * * * • 
is a series of functions of any number of variables which are continuous at 
a point P where the series is convergent, then it represents a (■ multi-valued ) 
function which is continuous at P, provided we can find a second series 
fli + U 2 + . . . . 
where the IPs are positive functions, which , as well as their sum, are continu- 
ous at P, and are such that, at all points of a certain neighbourhood of P, 
I u i I IR 
for all values of i. 
Theorem . — If u(x , y) is a continuous function of the ensemble (x , y) 
for the value y = y 0 , and u(x , y)dx is convergent for y=y 0 , then, if we 
can find a positive function U(x , y), continuous for y = y 0 , and such that 
J"U(x , y) is continuous at y = y 0 , while 
I u ( x > y) I <U(x, y) 
