150 Proceedings of the Royal Society of Edinburgh. [Sess. 
The points of the wedge in the last two hyperplanes, which may he called 
its basal hyperplanes, do not come into consideration, since m(x , y ; z , w) 
is undefined there. 
Let k be any value between the upper and lower bounds (not included) 
of m(x , y ; x, y') in the primary closed rectangle. Then we can find two 
finite values 1\ and k 2 , both of which are assumed by the double incre- 
mentary ratio, where 
k Y <k< h 2 . 
By what precedes there is a point P x of the four-dimensional wedge (5) 
at which 
m(x , y ; z , w) = k Y , 
and a point at which 
m{x , y ; z , to) = k 2 . 
If this second point is not on one and the same hyperplane boundary as 
P x , let it be P 2 . In the contrary case take as P 2 a neighbouring point 
inside the wedge at which 
m(x , y ; z , w) = k' 2 , 
where 
k k 2 , 
this is possible by reason of the continuity of m(x , y ; z , w). 
The stretch P 1 P 2 will then be such that every one of its internal points 
is internal to the wedge, so that mix , y ; z , w) is definite at every internal 
point of P 1 P 2 , and has at the end-points the definite values k x and k 2 (or k' 2 ). 
At every point of the closed stretch P^ , therefore, m(x , y ; z , w) is a 
continuous function of the ensemble (x , y , z , w), and therefore assumes 
at some point internal to P X P 2 , and therefore to the wedge, the value k 
intermediate between its values at the end-points P x and P 2 . This proves 
the theorem.* 
Cor . — The upper and lower bounds of the double incrementary ratio 
in the rectangle are unaltered if we omit any or all the boundary points 
from the possible positions of the points (x , y) and (x', y). 
§ 8. Returning to § 6, the result of Cor. 4 is sufficiently interesting to 
merit separate proof. 
Theorem 6. — If f x exists throughout a closed area and is a continuous 
function of y, the value of any derivate of f x with respect to y at any 
internal point P of the area lies between the upper and lower bounds of 
the derivate in a non-axial f neighbourhood of P. 
* The extension of this result to the case when f(x , y) is a quasi-finite function, which, 
in the case of one variable, was given in the paper quoted on p. 144, line 2, is not elaborated 
here, the attention being concentrated in this paper, in the first instance, on finite functions, 
t That is, omitting the axial cross x — a and y = b. 
