146 Proceedings of the Royal Society of Edinburgh. [Sess. 
where L and U are the lower and upper bounds of any derivate of f x with 
respect to y at points internal to the rectangle. 
For, since 
m(a, l ; a + h, b + k) - m <^ b + ^ ~ m “ ( * ’ 6 + *> , 
h 
and the conditions of the Theorem of the Mean are satisfied, 
m(a , b ; a + h , b + k) ^ ) — fx( x > b) w p ere a<x' <a + h , 
k 
= incrementary ratio of function f x (x, y) at the pair of 
values y = b, y = b + k. 
Hence 
Iv L w(a , b ; a + h , b + k) L\J X , 
where L x , and are the bounds of this incrementary ratio at pairs of 
values of y internal to the interval (b , b + k). Since these bounds are the 
same as those of any derivate of f x (x', y) with respect to y at points 
internal to the same interval (that is, at certain points internal to the 
rectangle), we have, a fortiori, 
L L m(a , b ; a + h , b + k) L U . Q.E.Lc 
Theorem 3 . — If f x exist in a closed neighbourhood of the point (a , b), 
and any derivate of f x with respect to y has only one double limit as we 
approach the point (a , b) by means of points not on the axial cross, then 
f x has a differential coefficient f yx with respect to y at the point (a , b). 
For, by Lemma 4, 
L L m(a , b ; a + h , b + k) L U , 
where L and U are the lower and upper bounds of the derivate in question 
in the completely open rectangle ( a,b ; a + h , b + k). 
But, under the given condition, L and U evidently have the same limit 
when h and k each approach zero in any manner whatever. Hence 
m(a , b ; a + h , b + k) has only one double limit, viz. the common limit of 
L and U. 
F urther, f x being defined on the ordinate x = a, the derivates of f x with 
respect to y are, by Lemma 3, double limits of m(a , b ; a + h , b + k). They 
are therefore all equal to the unique double limit of m(a , b ; a + h , b + k), 
so that f x has a differential coefficient at (a , b) with respect to y. 
Note. — I f f x exist on the ordinate x — a, and any derivate with 
respect to y of f x is continuous with respect to y at the point (a , b), 
then fyx exists at the point (a , b). . 
This follows, of course, from the well-known theorem in one 
dimension. 
