1908-9.] Reversibility of Order of Partial Differentiation. 149 
ensemble (x , y). If we make this assumption, the invariability of the 
bounds results at once from the simple fact that the double incrementary 
ratio assumes every value between its upper and lower bounds in the 
closed rectangle at pairs of points internal to the rectangle ; a proof of 
this is appended, similar to that given for one dimension in the paper on 
derivates quoted, and easily generalisable for ^-dimensions. 
The assumptions actually made so far, however, are only that f(x , y) 
should be continuous with respect to x , and f x with respect to y. From 
this it follows that f(x ,y) is a continuous function of x and of y (though 
not necessarily of the ensemble (x , y)), save for an arbitrary function of 
y alone, which disappears, being purely additive, from the double incre- 
mentary ratio. 
It will be found later that even the existence of f x is not essential ; it 
is sufficient if any one of the derivates of f(x , y) with respect to one 
variable is a continuous function of the other variable (§ 14). 
Theorem 5. — The double incrementary ratio 
mix ,y,x', y ) = f{x ’ ~/< x ’/> V) . g) 
(x-x)(y -y) 
where f(x , y) is a continuous and finite function of the ensemble (x,y), 
assumes every value between its upper and lower bounds in a closed 
rectangle (a , b ; c , d) at points at which 
a<x <x<b , and c< y' < y <cl . . . . ( 1 ) 
Consider the four-dimensional function m(x , y ; z , w). This is definite 
and continuous at every point of the closed four-dimensional parallelepiped 
(a , a , b , b ; c , c , cl , d), bounded by the eight hyperplanes 
x = a , x = c, y = a , y — c , z = b , z = d, w — b, w = d 
excepting along the diagonal hyperplanes 
Moreover, identically 
x = y,z = w 
m(x , y ; z , w) = m{y , x ; z , w)=- m(x , y ; w , z) 
( 2 ) 
(3) 
(4) 
so that all the values which are assumed by m(x , y ; z , w) in the parallele- 
piped, and on its boundary are the same as those assumed at points of the 
four-dimensional wedge 
a^x <x^fb , c^y' <y^d . . . . . (5) 
bounded by the six hyperplanes 
x = a, y = b , z — c, iv == d , and x = y , z — w 
( 6 ). 
