152 Proceedings of the Poval Society of Edinburgh. [Sess. 
This corollary, together with the one-dimensional theorem given in the 
note on Theorem 3, constitutes a proof of Theorem 3. (Cp. also the note 
at the end of Theorem 1. It is unnecessary to do more than call attention 
to the special case in which f x has a differential coefficient with respect to 
y throughout the non-axial neighbourhood.) 
§ 9. Theorem 7 . — If f(x , y) is a finite continuous function of x at every 
point of a closed rectangle (a , b ; a + h, b + k), and has, except possibly on 
the bounding ordinates x = a and x = a + h , a differential coefficient f x 
with respect to x, which is a finite continuous function of y : (1) there is 
a point internal to the rectangle at which one lower derivate of f x with 
respect to y \ m(a , b ; a + h , b + k), and the upper derivate on the other side 
Am(a,b;a + h, b + k); (2) each of the derivates of f x with respect to y 
assumes vahces A m(a , b ; a + h , b + k) and values L m(a , b ; a + h , b + k). 
For 
m(a, b ; a + h, b + k) = b + k ) fL m A b ’ h + 0 = p, a+m (b , b + k) 
h ax 
_f x (a + Oh , h + k) -f x (a + Oh , h) 
~k 
= incrementary ratio of f x (a + Oh , y) regarded as a function of y. 
Hence, by Theorem 4, p. 10, of the paper “ On Derivates and the Theorem 
of the Mean,” already quoted, the first result follows, and by Theorem 5 of 
the same paper we get the result (2). 
§ 10. We now return to the question touched on as to the definition of 
the second partial differential coefficients. So far we have followed the 
usual definition adopted by Schwarz and others. This postulates that 
ij- should only be considered to exist at a point (a ,b) if exists on the 
dxdy 17 iv/ gy 
abscissa y — b at and in the neighbourhood of the point (a , b), and similarly 
that should only be considered to exist at the point (a , b) if exists 
dy dx dx 
on the ordinate x~a at and in the neighbourhood of the point {a, b). 
This definition was historically inevitable. To speak of a second differ- 
ential coefficient implies that we are dealing with a differential coefficient 
of a differential coefficient. Extension of knowledge has, however, always 
led to an extension of meaning, and there is no a priori reason why we 
should not, if we find it convenient, extend the definition as follows : — 
df 
If exists at the point ( a , b) but not in the neighbourhood of (a , b) 
on jthe ordinate x — a, and yet it happens that the derivates with respect 
to x of fix , y) (upper and lower, left and right) should all four have a 
