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Proceedings of the Royal Society of Edinburgh. [Sess. 
m(x ,y; x' , y') in the closed rectangle is that in the open rectangle, and 
therefore, by (1), ^ the lower bound of derivate in the open rectangle. 
Cor. 2. — The bounds of the derivates of f x with respect to y in any 
rectangle (a , b ; a + h , b + k) are unaltered if we include the left-hand and 
the right-hand bounding ordinates of the rectangle , and the bounds of the 
right-hand ( left-hand ) derivates are unaltered if we include the lower 
(upper) bounding lines. 
For in any of these cases the argument used in proving (1) holds. Thus 
the result (1) still holds. The lower bound of the derivate having 
certainly not been increased by the introduction of the new points, 
(2) also holds, whence it follows that the lower bound of the derivate is 
still equal to that of the double incrementary ratio. 
The results of this theorem and its corollaries may be shortly summed 
up in the following general statement : — 
The bounds of any derivate of f x in any rectangle (a, b; a + h, b-f-k), 
open or closed, are the same as those of the double incrementary ratio in 
the same rectangle, including or not some or all of its boundary points. 
Here the expression “ derivate of f x in a rectangle ” is to be so under- 
stood that only such derivates are included at boundary points as result 
from operations in the rectangle ; e.g. at the left-hand bottom corner, only 
the right-hand derivates with respect to y of f x . 
Cor. 3. — The bounds of any derivate of f x and of the double incre- 
mentary ratio in any neighbourhood of a point P are unaltered if we 
omit the axial cross through P.* 
Cor. 4. — All the derivates of f x leave the same associated plane limiting 
functions f and \Js, and these are the same tuhether the values at the point 
itself be included or not, and are still the same if in calculating and \fs 
we omit the axial cross through P. 
In particular, therefore, all the derivates of f x lie between their <p and 
\f, and if one of these derivates is continuous with respect to the ensemble 
(x , y), so are they all, and they are all equal. Moreover, the same is true if 
it is only known that one of these derivates has a unique limit as we 
approach the point by points not lying on the axial cross. 
§ 7. The fact that the bounds of the double incrementary ratio in a 
rectangle (a ,b; a-\-li , b + k) are unaltered, when we include some or all of its 
boundary points, is at first sight remarkable, considering that we have not 
made the assumption that /( x , y) is continuous with respect to the 
* Or any isolated set of axial crosses ; that is, any set of axial crosses such that, taking 
any point P internal to the region considered, we can find a region containing P as internal 
point and intersected by at most one of the axial crosses. 
