140 
Proceedings of the Royal Society of Edinburgh. [Sess. 
internal or end-point, we get in this manner a constantly increasing set 
of integers m 1 , m 2 , . and another n 1} n 2 , . . . . such that Q is the 
sole limiting point of the points P TO . The corresponding series F(m*,?R) 
has then u for limit, which shows that u is a double limit of F (m,n). 
Q.E.D. 
If a finite number of the quantities v(n) are infinite, we can omit them 
from consideration. If, however, this is not the case, u is + oo or — oo and 
is equal to all but a finite number of the v(n)’a. In this trivial case it is 
clear that, corresponding to any v(n) = u, we can find an integer m such 
that F(m , n) is numerically greater than any chosen quantity, and in this 
way we can construct a series F(m< , nf having u as limit. 
Thus in any case the theorem is demonstrated. 
Lemma 2 . — If f(x , y) has only one double limit at a point (a , b), and 
the simple limit 
L t f(a + h, b + k) 
h = 0 
is unique for all values of k in a certain neighbourhood of the zero point , 
then the repeated limit 
L t f(a + ii , b + k) 
ft=0 7i=0 
exists. 
This is an immediate consequence of Lemma 1. 
N.B . — If the simple limit is not unique, it follows from Lemma 1 that, 
however we define the limiting function as a function of y, all its limits 
are double limits of f(x ,y). Hence if, as in Lemma 2, f(a + h ,b + lc) 
has only one double limit, all the limits of all possible limiting func- 
tions coincide. In particular the upper upper and lower lower limits are 
equal. 
Note 1. — It may evidently happen that even when L tf(x,y) does 
x=a 
not exist as a unique limit, the upper and lower limits, and therefore 
all intermediate limits, have one and the same definite limit as y 
approaches b. If we agree with some writers to call this the repeated 
limit, it is evident that Lemma 2 still holds, omitting the assumption 
as to the uniqueness of the simple limit. 
Note 2. — The existence of a unique limit of f(x,y) for each fixed 
value of x does not, of course, involve the existence of a unique limit 
when y varies with x, which would be concomitant to the existence of 
a unique double limit. 
§ 3. A rectangle whose axes are parallel to the axes of coordinates, 
and whose lower left-hand and upper right-hand corners are respec- 
