164 
Proceedings of the Royal Society of Edinburgh. [Sess. 
ensemble (x , y) at the point (a , b), and f xy and f yx both exist at the point 
(a , b), then 
fxy(a , b) =f yx (a , b) , 
and m(a, b; a + li, b + k) has a unique double limit equal to both these 
repeated partial differential coefficients. 
For, since f xx is a continuous function of (x , y), and f yx exists at the point 
(i a , b), it follows by the preceding theorem that m(a , b ; a + h . , b + k) has a 
k 
unique limit for all modes of approach for which j has not the limit 
h 
zero, and this limit is f yx (a , b). 
Again, since f yv is a continuous function of (x , y), and f xy exists at the 
point (a, b ), it follows that m(a , b ; a + h, /> + /»;) has a unique double limit 
h 
for all modes of approach for which 
k 
has not the limit zero, and that 
this limit is f xy {a , b). 
Hence f yx (a , b) = f xy (a, b) — unique limit of m(a , b; a + h , b + k) for all 
modes of approach which do not make either 
But for modes of approach which make 
k 
h 
h 
or 
k 
k 
h 
have the limit zero, 
have zero for a limit. 
m(a , b ; a + h , b + k) has, by the above, the limit f xy (a , b), and for modes 
of approach which make ~ have the limit zero, it has f yx (cc , b) for limit. 
Hence for all modes of approach m(a , b ; a + h , b+k ) lias the unique limit 
fja, b) = f, s (a, b). 
( Issued separately March 2 , 1909 .) 
