1908-9.] Reversibility of Order of Partial Differentiation. 157 
Hence we see that the various derivates of the function mfib , b + k ) at the 
point x = a are the various limits of m(a , b; a + h , b + k) with respect to 
h , k being constant. 
There are two special sequences of values of h on the right which need 
consideration. The first is that which yields the upper derivate with 
respect to x of f(a, b + k), say f +x (a , b-\-k), as the limit of 
f(a + h, b + Jc) -f(a , b + k) 
This will yield a limit, or limits, of 
f(a + h , b) -f(a , b) 
• _ 
not greater than the upper derivate f +x (a , b), and therefore for 
m(a , b; a + h , b + k) a limit, or limits, not less than 
f +x (a , b + k) -f +x (a , b) ' 
k 
Hence, by what was said above, this last expression is L the upper 
right-hand derivate of m x (b , b + k). 
The second sequence is that which yields f +x (a , b). The consideration 
of this sequence shows in like manner that the expression (3) is the 
lower right-hand derivate of m x (b , b + k). 
Thus the expression (3) lies between the upper and lower right-hand 
derivates of m x (b , b + k). 
Similarly, considering the sequences which yield respectively the lower 
right-hand derivates, we find that the incrementary ratio of the lower 
right-hand derivate, that is, 
/+*(«, b + k) -f +x (a, b) 
k 
lies between the upper and lower right-hand derivates of m x (b , b + k). 
Similarly, the incrementary ratios of the left-hand derivates of f(x , y ) 
lie between those of m x (b , b + k). 
§ 14. Theorem 8 . — If f(x , y) is a finite continuous function of x at 
every point of a closed rectangle (a , b ; a + h , b + k), while any one of the 
four derivates (upper, lower, left , and right) of f(x , y) with respect to x is 
a finite continuous function of y , except possibly on the bounding 
ordinates, then any derivate with respect to y of this derivate with respect 
to x assumes values both L and also A the double incrementary ratio 
m(a , b ; a + h , b -f- k) of the rectangle at points internal to the rectangle. 
For m x (b , b + k) is then also finite and continuous with respect to x in 
