1908-9.] Reversibility of Order of Partial Differentiation. 151 
Further , the upper and lower bounds in the non-axial neighbourhood 
are the same as those in the closed neighbourhood of P. 
For 
m( a , b; a + h, b + k) = ’ 6 + / ' :) ’ '' + ^ = hn a+eh (b ,b + t) 
k ax 
f x (a + 6h, b + k) -ffa + Oh , b) 
h 
where 0<(9< 1. 
Now this latter single incrementary ratio lies between the upper and 
lower bounds of the derivates of f x (a + Oh , b + 0'k), where 0<d , <l, Oh 
being constant ; a fortiori, between the bounds of the same when Oh is 
variable ; and again, a fortiori, between the upper and lower bounds of any 
derivate of f x at all points {x' , y') of the neighbourhood not on the axial 
cross. 
This being so, it is true of the unique limit of m(a , b] a + h , b -f k) with 
to h, that is, of ' ^ a ’ ■ ^ ^ 5 and therefore of all the limits 
/c 
of this last incrementary ratio, that is, the derivates of f x at the point (a , b). 
To prove the second part of the theorem, we only have to notice that 
the value of any such derivate of f x on the axial cross being a limit of 
values in the non-axial neighbourhood, lies between the bounds in that 
neighbourhood. 
Cor. 1. — If <fi and \js are the associated plane limiting functions of any 
derivate of f x , that derivate lies between <p and \js at every point, and f 
and \h are the same whether defined in Bairds * manner or that usually 
adopjted by myself f or calculated always with reference to a non-axial 
neighbourhood. 
It should be noticed that the corresponding results for the first deri- 
vates, obtained from one-dimensional theory, are less extended. Any deri- 
vate with respect to x of f(x , p) , when fix , y) is continuous with respect 
to x throughout a closed area, lies between its upper and lower bounds in 
any neighbourhood omitting the ordinate of the point. Hence the omission 
of the ordinate does not affect the bounds, and the plane associated 
functions are the same whether defined as by Baire, or in my usual 
manner, or calculated from neighbourhoods omitting the ordinate. 
Cor. 2. — If any derivate of f x has only one limit in a non-axial 
neighbourhood of a point P, it is continuous at P with respect to the 
. ensemble (x , y) , and therefore in particular is continuous at P with 
respect to y. 
* Closed neighbourhood. 
t Open neighbourhood, omitting values at the point itself. 
