1908-9.] Reversibility of Order of Partial Differentiation. 143 
zero as limit. Hence, a repeated limit being a double limit (Lemma 1), all 
the repeated limits of m(a , b; a + h, b + k) are equal. 
But, by hypothesis,/^. is defined on the ordinate x = a, so that 
JA a ' = YJ, m(a ,b)a + h , b + k ) , 
k h = 0 
and has, therefore, by what has been shown, only one limit when k has 
zero as limit ; that is to say, f x has a differential coefficient f yx with respect 
to y at the point (a , b). 
Note. — It should be noticed that it does not follow that f yx is con- 
tinuous at (a , b) with respect to either variable, still less with respect 
to the ensemble ( x , y). In fact, f yx need not exist on the axial cross, 
except at the point (a, b) itself. It will be proved below (§8, 
Theorem 7, Cor. 2), however, that when it does exist on the axial 
cross, it is continuous at (a, b). 
Theorem 2 (the Dini-Schwarz Theorem). — If in addition to the require- 
ments of the 'preceding theorem , f y exists along the line y — b at and in the 
neighbourhood of the point (a, b), then f xy also exists at the point (a, b) 
and has the same value as f yx . 
For, in this case, 
f y (a + h , b) - f y (a , b) 
h 
U m(a , b ; a + h , b + k) , 
k = 0 
and has, therefore, as was shown in the preceding proof, only one limit 
when h has zero as limit ; that is to say, f y has a differential coefficient f xv 
with respect to x at the point (a , 6). 
Since the value of f xy (a , b), like that of f yx (a , b), is thus the unique 
double limit of m(a , b; a + h , b-\-k), 
fxy(<* , b) =f yx {a , b) . Q.E.D. 
Note 1.— -It has nowhere been assumed that the unique limit postu- 
lated is finite ; it may be + oo or — oo . 
Note 2. — The arguments used in proving the Repeated Theorem of 
the Mean, as well as Theorems 1 and 2, which depend on it, being based 
on the Theorem of the Mean in one dimension, do not require the full 
assumption that f yx exists at every point internal to the rectangle. 
It suffices, in fact , if there is no distinction of right and left * with 
respect to the derivates of f x regarded as a function of y, as follows 
* Thinking of the representation in two dimensions, right and left with respect to y is, 
of course, “ up and down.” - i 
