1908-9.] Reversibility of Order of Partial Differentiation. 163 
Since this is true for all positive values of m , it is true for all modes of 
approach such that j has not zero for one of its limits, which is equivalent 
to the statement to be proved. 
Theorem 10 . — If f xx is a continuous function of the ensemble (x , y) at 
the point (a , b), and f yx exists at the point (a, b), then the double in- 
crementary ratio has a unique double limit whose value is f yx (a , b) for 
k 
all modes of approach other than those for which 
limits. 
For 
where 
Therefore 
has zero for one of its 
m 
(a, b; a + h, b + Tc) = > h - m ^ b ’ b + k \ 
/ v 
mfb , b + k) = 
f(r , b-\-k) — f(x^ b) 
d 
m(a , b ; a + h , b + k) = —mfb , h 4- li) for some value of x> a and <a +-h , 
(- O /y. 
where 
_ fJl^ d* Oh , h + k) —f x ((i + Oh , ?>) 
Q<0<\ , 
(D 
and 0 is an otherwise unknown function of h and k. 
But since, by hypothesis, ~f x is a continuous function of the ensemble 
(x , y ) at the point (a , b), and ~f x exists there, we may apply the lemma. 
d„ 
But if 
has not zero for one of its limits, 
A 
Oh 
certainly has not zero 
for one of its limits, since, whatever function 0 is of li and k , 
Oh 
> 
k 
h 
Hence the lemma holding for the function ffx , y) 
ffei + Oh ' , b + It) — f fa + Oh , b) 
has a unique limit, and this is f y fa, b). Hence, by (1) the same is true 
of m(a , b ; a + h, b + Jc) for all the specified modes of approach of h and k 
to zero. 
Theorem 12 . — If f xx and f yy are both continuous functions of the 
