158 Proceedings of the Royal Society of Edinburgh. [Sess. 
the closed ^-interval {a , a + h). We may therefore apply Theorem 5 of 
the paper “ On Derivates,” already quoted. Hence any derivate of 
m x (b , b-\-k) assumes at points internal to this ^-interval values both L 
and also A m(a , b; a + h , b + k), since this is the incrementary ratio of 
mfb , b + k) for the pair of ^-values a and a + h. But, by the preceding 
article, the incrementary ratio of the upper right-hand derivate 
f +x (x , b + lc) -f +x (x, b) 
~k 
lies between the upper and lower right - hand derivates of m x (b , b-\-k) 
at every point, and therefore also assumes values both L and also A 
m(a , b ; a -j- h , b -f- k). 
Since, by the preceding article, a similar argument applies to any other 
derivate of fix , y) with respect to x, this proves the theorem. 
Theorem 9. — Under the same assumptions as in the preceding theorem, 
the upper and lower bounds of the derivates with respect to y of the 
chosen derivate with respect to x in the completely open rectangle 
(a , b ; a + h , b -f k) are the same as those of the double incrementary 
ratio m(x , y ; x 7 , y'J. 
For, if L' and U 7 denote the lower and upper bounds of m(x , y ; x, y'), 
it follows from the preceding theorem that the repeated derivate considered 
assumes values A any quantity <XJ / and values L any quantity <L'. 
Hence 
the lower bound of the repeated derivate L If L IT L upper bound of repeated 
derivate . . (1) 
But, by § 10, the repeated derivate in question lies at any point (x , y) 
internal to the rectangle between the bounds of mix , y ; x', y'), when the 
point ix ! , y') approaches (x , y) as limit. Therefore the repeated derivate 
at the point (x , y) lies between quantities which themselves lie between 
1/ and U', so that L' L chosen derivate at (x , y) L U 7 , and therefore 
L' L lower bound of repeated derivate L upper bound of same L U' . (2) 
from (1) and (2) If is the lower and IT 7 the upper bound of the repeated 
derivate. Q.E.D. 
Cor. 1 . — The bounds of the double incrementary ratio in any rectangle 
(a,b; a + h , b + k) are unaltered if we include in the rectangle any or 
all of its boundary points. 
For the argument used in proving (1) is equally valid if 1/ and JJ' 
denote the bounds of mix , y ; x', y') in the closed rectangle. Also, since 
this change does not increase L 7 nor decrease U 7 , (2) still holds ; hence the 
