154 Proceedings of the Royal Society of Edinburgh. [Sess. 
proving that all the derivates with respect to y of the upper right-hand 
derivate with respect to x are equal at the point (a , b). F or definiteness 
the upper right-hand derivate was taken, but it is clear that the same 
proof applies for any derivate. Thus, without assuming the existence of 
fx on the - ordinate x — a except at the point (a , b), all the derivates with 
respect to y of all the derivates with respect to x of fix , y) are equal at 
the point (a , b), so that, in Hobson’s sense, the partial differential co- 
efficient 7 J - exists at the point (a , b). 
dydx 
Adopting Hobson’s definition, the set of sufficient conditions for the 
reversibility of the order of partial differentiation discussed by Schwarz, 
Dini, and Hobson, reduce, in the light of what proceeds, to the following 
simpler form : — 
d 2 f 
It (D d p- exists at all ‘points in a non-axial neighbourhood of the 
point (a , b) ; 
d'2f 
(2) The values of in a non-axial neighbourhood of the point 
(a , b) have a unique limit at (a , b), finite or infinite ; 
df .... 
(3) — not only , as 'is implied in (1), exists, and is a finite and con- 
UX 
tinuous function of y in a non-axial neighbourhood of (a , b), but also 
exists on the abscisse y — b , cd and in the neighbourhood of (a , b) ; 
df 
(4) — exists at the point (a , b) ; 
dy 
then, in Hobson's sense, 
d 2 f 
and 
d 2 f 
dxdy 
both exist and are equal to the 
dydx 
unique limit specified in (2). 
§ 11. It should, however, be noticed that the retention of the existence 
of the first differential coefficient in Dr Hobson’s definition of the second 
differential coefficient must be regarded, from the point of view adopted by 
him, as somewhat arbitrary. It seems not less unreasonable to assert the 
existence of 
d 2 f 
dydx 
when all the derivates with respect to y of all the 
derivates with respect to x coincide at the point (a , b) , which may, of 
df 
course, be the case without f- existing at (a , b ) . 
ax 
Adopting this definition, if we know that the double incrementary 
ratio has a unique limit at the point (a, b ) , we can easily show that all 
the derivates of derivates of f{x , y) are equal at (a, b). For, although the 
derivates of derivates are not necessarily themselves repeated limits of 
