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Proceedings of the Royal Society of Edinburgh. [Sess. 
a differential coefficient f x with respect to x, which is itself a finite con- 
tinuous function of y with, at every internal point of the rectangle, a 
differential coefficient 
df _ df 
dx dydx ’ 
then there is an internal point of the rectangle at which the repeated 
differential coefficient f yx (x' , yf is equal to the double incrementary ratio 
of the rectangle, viz. 
m(a, b; a + h, b + k)=f yx (x , y) . 
For, applying the Theorem of the Mean to the identity (3), we get 
m(a , A; a + h, b + k) = ~m x (b, b + k) at some point x between a and (a + h) not 
inclusive 
_ fffi , b + k) — f fix , b) 
k 
whence the result follows by a second application of the Theorem of the 
Mean, since f x is continuous with respect to x on the sides y = b and 
y = b + k of the rectangle. 
Note. — Assuming the simple result that, when f(x, y) is a con- 
tinuous function of the ensemble (x , y) the double incrementary 
ratio assumes at points internal to a rectangle (a, b ; a + h , b + k ) every 
value between its upper and lower bounds,* this shows that if f yx then 
exists and is finite at every internal point of this rectangle it also 
assumes every value between its upper and lower bounds. 
§ 4. Theorem 1 (on the existence of f yx at the point (a , b)). — If f x 
exists in a closed neighbourhood of a point (a , b), while in the completely 
open neighbourhood, excluding the axial cross, j* it has a differential co- 
efficient f yx with respect to y, then, if f yx has only one double limit as we 
approach the point (a , b) in any manner by means of points not on the 
axial cross, f yx exists also at the point (a , b) itself. 
For, by the Repeated Theorem of the Mean, 
m(a , b ; a + h, b + k) =f yx (x , y) , 
where the point (x ' , y') does not lie on the axial cross, and has (a , b) as 
limiting point when h and k each approach zero in any manner without 
assuming the value zero. 
Since f y fix , y), and therefore f y fix , y'), has only one double limit at the 
point (a, b), the same is true of m(a , b; a + h, b + k) when h and k have 
* A proof of this is given in § 7 below, 
t That is, not on x = a nor on y = b. 
