1908-9.] Reversibility of Order of Partial Differentiation. 159 
bounds of m(x , y ; x. y') in the closed rectangle are the same as those of 
the repeated derivate in the open one, and therefore are the same as the 
bounds of m(x , y ; x' , y') in the open rectangle. 
Cor. 2. — The bounds of a repeated derivate in a rectangle 
(a , b ; a 4- h , b + k) are unaltered by the inclusion of boundary points, in 
as far as at each such boundary point the repeated derivate in question 
is defined by means of sequences of points lying in the rectangle. 
Thus, for instance, in dealing with the right-hand derivates with 
respect to y of the right-hand derivates with respect to x, we may include 
the left-hand bottom corner ; and in dealing with the right-hand derivates 
with respect to y of the left-hand derivates with respect to x, we may 
include the right-hand bottom corner. 
For in this case the values of h and k used in the reasoning of § 10 
do not take us out of the rectangle, so that (2) still holds, the bounds both 
of the derivate and the double incrementary ratio being taken with 
respect to the rectangle including the boundary points in question. Since, 
by the preceding corollary, the bounds of the double incrementary ratio are 
the same as in the open rectangle, (1) also still holds, since the introduction 
of new points does not increase the lower bound or decrease the upper 
bound of the repeated derivate. 
Hence the bounds of the repeated derivate are the same as those of 
the double incrementary ratio, and are therefore unaltered. 
Cor. 3. — The bounds of any repeated derivate of f(x , y) and of the 
double incrementary ratio are unaltered, if we omit the axial cross 
through P. 
Cor. 4. — All the derivates with respect to y of any derivate of f(x , y) 
with respect to x, which is a continuous function of y in a neighbourhood 
of a point P , have at P the same associated plane-limiting functions 
<p and \Js , and these are the same whether the values at the point itself be 
included or not, and are still the same if in calculating <p and \Js we omit 
the axial cross through P. In particular, therefore, all the derivates of 
such derivates lie between their cf and \fs , and if one of these repeated^ 
derivates is continuous with respect to the ensemble (x , y), so are they all, 
and they are equal. Moreover, the same is true if it is only known that 
one of these repeated derivates has a unique limit as we approach the 
point by points not lying on the axial cross. 
§ 15. It should be noticed that the restriction in the preceding theorem 
and corollaries, implied in the fact that we only consider derivates with 
respect to one variable of a function or derivate which is itself a 
continuous function of that variable, is an essential one. In fact, the 
