1908-9.] Reversibility of Order of Partial Differentiation. 153 
differential coefficient with respect to y, and, farther, all these differential 
coefficients are equal, this common value shall be called the partial differ- 
ential coefficient — =- . 
dyax 
Hobson, who has recently adopted this definition, * is thus ei abled to 
elf 
omit from the Schwarz- Dini conditions the condition that -A should exist 
ay 
on the abscissa y = b, except at the point (a , b ) itself. 
It is evident from § 4, Note 4, of the present paper that with this 
definition we must, if we are to be consistent, omit the condition included 
in that specifically retained by Hobson in a gloss on his condition (l),j* 
that ^ exists in the neighbourhood of the point (a , b) on the ordinate x = a. 
In fact, the assumption that A exists at the point ( a , b) is equivalent 
to the statement that, whatever sequence of values with zero as limit we 
ascribe to h, 
f(a 4- h , ,b) —f(a , b) 
JT 
has the same limit. 
At any point (a, b-\-k) let us take a sequence of values of h, giving the 
upper right-hand derivate of f(x , y) with respect to x as the unique limit of 
/(a + A- . n ; b + k) —/(a + h kiU , b) 
bjc , n 
At the point (a , b), however, any sequence of values of li will do, by the 
assumption made. Hence the incrementary ratio of this derivate may be 
written 
+ b + k)-f(a + h ktn , b) _ u f(a + h k>n , b)-f{a, b) 
n=co h k _ n n==cc b k n 
= U m(a , b ; a + h kn , b + k) . 
go 
Thus the derivates with respect to y of the upper right-hand derivate of 
f{x , y) with respect to x are repeated limits, and therefore double limits, of 
the double incrementary ratio, so that they all coincide at {a , b) if a unique 
double limit exists. 
df 
As pointed out in Note 4, § 4, the existence of -A on the ordinate x = a 
CLOG 
is immaterial in the proof of the existence of a unique limit for the double 
incrementary ratio. Thus the assumption in question is superfluous in 
* Loc. cit. 
t Line 13, loc. cit. 
