136 
Proceedings of the Royal Society of Edinburgh. [Sess. 
IX. — On the Conditions for the Reversibility of the Order of 
Partial Differentiation. By W. H. Young, Sc.D., F.R.S. 
(Communicated by J. H. Maclagan Wedderburn, D.Sc.) 
(MS. received November 4, 1908. Read February 1, 1909.) 
§ 1. In the case of a function of two variables f(x , y) there is, in general, 
no connection between the results of proceeding to the various limits, 
upper, lower, and intermediate, first with respect to x and then with 
respect to y, and first with respect to y and then with respect to x. We 
cannot even assert that the repeated upper limit obtained in one way is 
not less than the repeated lower limit obtained in the other way. 
The so-called necessary and sufficient conditions for the equality of 
the two repeated limits do little more than express in e-language the fact 
of the equality. There is, however, one simple case in which we can 
assert with confidence the existence of the repeated limit, this is, when a 
unique double limit exists. 
It is no accident, therefore, that the simplest and best-known set of 
conditions for the reversibility of the order of partial differentiation — - 
that associated with the names of Dini and Schwarz — is based on this 
property. 
The main object of the first of the three parts of the present paper is to 
obtain these Schwarz-Dini conditions in a more precise form than has yet 
been given to them. It is found that the axial cross through the point (a , b) 
considered may be omitted in obtaining the limit of the values of ~ in 
17 & dy dx 
the neighbourhood of the point (a , b), and indeed that the existence of 
d_ df 
dy ' dx 
except at the point (a , b) itself. 
It is hoped that the mode of proof adopted, in which the e-method is 
avoided, will be found somewhat simpler and more concise than the usual 
mode of presentation. 
In the second part of the paper I obtain certain properties of the 
repeated partial differential coefficients, or, more generally, of the derivates 
with respect to one variable of the differential coefficient, or of the 
derivates, with respect to the other variable. I am thus led to state 
on the axial cross is nowhere required for the purposes of the proof, 
