1908-9.] Reversibility of Order of Partial Differentiation. 137 
and prove certain extensions of the Schwarz-Dini conditions considerably 
less narrow than those previously given. 
In the third part of the paper I obtain another set of sufficient con- 
ditions which I believe to be hew both in the form given to them and in 
the mode of proof. These are as follows : — 
If both ^4 a nd ^4 are continuous functions of the ensemble (x , y) at 
the point (a , b) and both . 4^- and - 4 - . 4^- exist at the point (a , b), 
dy dx dx dy 
then these two latter repeated partial differential coefficients are equal 
at (a , b). 
Stolz in his Grundzilge der Differenzial- und Integralrechnung, vol. i. 
pp. 141-147, has given a set of conditions, also due to Schwarz, included 
in mine as a very special case. The additional demands made by Schwarz 
on the function consist of (1) the existence of ~ and ( j . yt i 
m a 
dy ' dx dx dy 
closed neighbourhood of the point (a, b), and (2) the continuity of 
f . uf- with respect to y and of ~ . rj- with respect to x at the point 
dy dx dx dy 
(a , b). 
The interesting question then arises : What is the relation of these 
conditions to those for the equality of two repeated limits ? The method 
of proof adopted shows that in this case also the equality virtually depends 
on the existence of a unique double limit. 
It may be noted also that in their relation to the repeated partial 
differential coefficients ~ . yf- and ~ . q— , or to the repeated derivates, 
dy dx dx dy 
the new conditions are less narrow than the Schwarz-Dini conditions. 
They do not demand even the continuity of the derivates of derivates at 
the point, still less the existence and continuity of ~ . in the neighbour- 
hood of the point considered. 
dy ' dx 
PART I. 
§ 2. With the object of rendering what follows as complete in itself as 
possible, we begin by formulating a few properties in the theory of limits, 
on the proper grasp of which the understanding of the subject hinges. 
We have first to explain exactly what we mean by a double and a repeated 
limit. 
x 1 , x 2 , .... x n , he a series of values of a variable x, forming 
