1908-9.] Reversibility of Order of Partial Differentiation. 147 
§ 6. I now proceed to give the property which may be regarded as con- 
stituting the raison d'etre of the sets of sufficient conditions for the 
reversibility of the order of partial differentiation with which we are at 
present occupied, in the case when f x and f y exist throughout an area. 
This property is that all the derivates of f x with respect to y, and all the 
derivates of f y with respect to x , have the same upper and lower bounds in 
every completely open area , and these bounds are the same as those of the 
double incrementary ratio in the same area. This follows at once from 
the following theorem, which results immediately from Lemmas 3 and 4. 
Theorem 4 . — The upper and lower bounds of cdl the derivates of f x at 
all the internal points of a closed rectangle with sides parallel to the axes, 
throughout which f x exists, and at every internal point of which f x is a 
continuous function of y, are the same and are equal to those of the double 
incrementary ratio m(x , y ; x', y') at pairs of points internal to that 
rectangle. 
For, if (a, b) is internal to the rectangle, so is a portion of its ordinate. 
Hence any derivate of f x with respect to y at {a , b), being a repeated limit 
of m(a ,b ; a-\-h ,b-\-k) , A lower bound of mix , y ; x' , y'), since we can 
restrict h and k so that the point (a + h, b + k) is also internal to the 
rectangle considered. Therefore 
lower bound of derivate A lower bound of m(x , y ; x, y ) . . (1). 
But, by Lemma 4, 
lower bound of derivate L m(x , y ; x, y) 
when the points (x , y), and (x', y') and the rectangle (x , y ; x' , y'), are 
internal to the region. Therefore 
lower bound of derivate L lower bound of m{x, y ; x, y) . . (2). 
From (1) and (2) the equality of the lower bounds of the derivate and the 
double incrementary ratio follows. Similarly, the equality of the upper 
bounds may be proved. 
Cor. 1. — The bounds of the double incrementary ratio in any 
rectangle (a , b ; a + h , b + k) are unaltered if we include in the rectangle 
any or all of its boundary points. 
For the argument used in proving (2) is equally valid if the points 
(x , y) and {x r , y') are on the boundary. Hence 
lower bound of derivate in the open rectangle L lower bound of mix , y ; x , y) 
in the closed rectangle. 
But we must take the sign of equality, since the lower bound of 
