144 Proceedings of the Royal Society of Edinburgh. [Sess. 
from the more general statement of the Theorem of the Mean given in 
the paper “ On Derivates and the Theorem of the Mean,” by W. H. 
and G. Chisholm Young, Quart. Journal of Math., Oct. 1908. 
Note 3. — It should be noticed that, without making any properly 
two-dimensional hypothesis, we can prove that f yx exists at the point 
(a , b) if we postulate that f yx exists and is finite along the ordinate 
x = a in some open neighbourhood of the point (a , b) but not at the 
point itself, and that it has a unique limit as we approach the point 
(a , b) along that ordinate. 
This is an immediate consequence of the Theorem of the Mean 
for a single variable applied to f x regarded as a function of y at 
points of the ordinate x = a. 
For this reason, as well as from the fact that the axial cross can, 
as we have seen, be omitted, the usual statements of the Schwarz-Dini 
conditions seem to leave something to be desired.* 
It may further be remarked that in Note 2 it is sufficient if the 
derivates of f x with respect to y on the ordinate x = a present no dis- 
tinction of right and left.f All these sets of conditions are, of course, 
sufficient but not necessary. It is obvious that sufficient conditions 
of a less restricted character can be formulated ; some of these will be 
found below. 
Note 4. — In the proof of the existence of a unique double limit 
for m(a , b\ a + h , b + k), the assumption (1) that f x exists in the open 
neighbourhood of the point (a, b), excluding the ordinate x — a, was 
rendered necessary in order to apply the Repeated Theorem of the 
Mean. 
This being postulated, the further assumption (2) that f x also 
exists on the ordinate x = a, at and in the neighbourhood of the point 
(a , b), is needed in order to ascribe a meaning to the expression 
fjp , b + k) - f x {ci , b) 
and so to prove the existence of f yx at the point (a , b). 
Similarly, without postulating (2), the existence of f xy requires the 
assumption (3) that f y exists on the line y = b, at and in the neighbour- 
hood of the point (a , b). 
* Stolz, Grundziige der Differencial- und Integralrechn ung, 1893, p. 147. Hobson, 
“Partial Differential Coefficients and Repeated Limits,” Proc. L.M.S. , 1906, series ii., 
vol. v. p. 234. See also Functions of a Real Variable , p. 318 and the errata. It should be 
noticed that the account in the book is really earlier than that in the paper, 
t See footnote on preceding page. 
