1908-9.] Reversibility of Order of Partial Differentiation. 141 
lively the points (a, b), (a-\-h ,b-\-k), will be referred to as the rectangle 
( a,b ; a + h , b + k), and its double incrementary ratio defined as follows: — 
m(a, b; a + h, b + /,) = f( a + h ’ h + k ) ~/( a + h ’ / ' i • b + k > + r '< a ’ Q , (1) 
till 
Here h and k may have any positive or negative values. 
Using m x (b , b + /<:) for the single incrementary ratio of f(x , y) when x is 
constant, 
mfb , b + k) = 
f(x , b + k) -fix , b) 
k 
we clearly have the following identity : — 
rn(a , b ; a + h,b + k) = m "+V’ A + > 6 + *> 
( 2 ) 
(3) 
so m(a, b; a + h, b + k) is the incrementary ratio with respect to x of 
the incrementary ratio with respect to y of f(x , y), and similarly it is the 
incrementary ratio with respect to y of the incrementary ratio with 
respect to x. 
If in (2), keeping x constant, we let k approach zero in any manner, 
and there is only one limit, this is, by definition, the partial differential 
coefficient with respect to y, f y (x , b), or C - . Forming the single incrementary 
f 
ratio of ffx , b) for the pair of values x — a and x — a-\-h, if this has a unique 
limit as h approaches zero, this is, by definition, the repeated differential 
coefficient with respect first to y and then to x, and will be denoted b y/ or 
d df > dff 
dx dy dxdy 
If f y exists, but its single incrementary ratio has more than one limit, 
the various limits are the various derivates at the point ( a , b ) of f y , and 
may be called the repeated derivates of f(x , y) first with respect to y and 
then to x. 
If f y does not exist, the upper and lower left- and right-hand derivates 
which appear instead form functions whose derivates with respect to x 
may be treated as the repeated derivates of f(x , y) first with respect to y 
and then to x. 
The following theorem will for convenience be referred to as the 
Repeated Theorem of the Mean. 
The Repeated Theorem of the Mean. 
If f(x , y) is a finite continuous function of x at every point of the 
closed rectangle (a , b ; a + h , b + k) , and has , except possibly on the bound- 
ing ordinates, 
x—a and x — a + h , 
