1908-9.] Reversibility of Order of Partial Differentiation. 139 
a series of values and at (a , 6) a corresponding set of limits.* If one of 
these is A (which may be + oo or — oc ), then A is said to be a double limit 
of f (x , y) for the ensemble of values (a , b). 
If the variables x and y are continuous, we may regard the ensemble 
(x , y ) as a point of the plane. The points (x 1 , yf) , (x 2 , y 2 ) , . then 
form a sequence with the point (a , b) as limiting point. They do not, 
however, form the most general such sequence, since the individual points 
are explicitly excluded from lying on the axial cross 
x — a , 
y = b, 
through the limiting point.f 
Lemma 1 . — Any repeated limit is a double limit. 
For convenience we shall use F(m, n) for f(pc m , y„). 
By the definition of a repeated limit there is then a sequence of values 
of F(m , n), keeping n constant, having a unique limit v(n), when m is 
indefinitely increased ; and the quantities v(n), as n is indefinitely increased, 
have the repeated limit, say u, as one of their limits ; that is 
v(n) = L t F (m , n) 
m = oo 
w = one of the L It v(n). 
n=m 
First consider the quantities v(n) to be all finite. 
Represent the values of the various functions on a straight line, 
/(m , n) by the point P m>n , 
v(n ) by the point Q n , 
u by the point Q . 
Then, under the given conditions, Q is the limiting point of the sequence 
Q x , Q 2 , • • • • and is, for each value of n, the limiting point of the 
sequence P 1>n , P 2> „, . . . . 
Since all these points, except possibly Q, are finite points, any interval 
d containing Q as internal or end-point determines a point Q n where n is 
the first integer greater than some chosen number such that Q„ lies inside d. 
The interval d then determines a point P m>n such that n is the first integer 
greater than a chosen number, and such that P m n lies inside d. 
Taking a sequence of such intervals d x , d 2 having Q as sole common 
* These may be regarded as falling under the heading of simple limits, since they are 
hit F (n), where F (n) =/(&„, y n ). 
n—oz 
t It is unnecessary here to enter into the modifications necessary when the point (a , b ) 
is at infinity. 
