156 Proceedings of the Poyal Society of Edinburgh. [Sess. 
everywhere * on the tc-axis, for each such value of x this derivate, regarded 
as a function of y , has its derivates undefined. We might, of course, agree 
still to define its derivates as having the value zero, since a change in the 
value of y makes no change in that of the function. In this case the above 
result would hold in a more extended sense. 
§ 12. From the inequality of the preceding article it follows that if 
m(a , h ; a + h,b + k) has a unique double limit, all the derivates of its 
derivates coincide at the point (a ,b)\ so that, in the sense indicated at the 
S}f d?f 
beginning of this article, we might choose to say that J and J both 
(JL1J CtJU CLJCCtlJ 
exist at (a , b) and are equal. 
In this sense the conditions for this to be the case, obtained by omitting 
superfluous conditions from the Schwarz-Dini conditions, are as follows : — 
d 2 f 
( 1 ) 
(a, b); 
dyd 
x 
exists at all points in a non-axial neigh bourhood of the point 
d 2 f • 
(2) The values of ^ ^ in a non-axial neighbourhood of the point (a , b) 
have a unique limit at (a , b), finite or infinite ; 
df 
(3) — exists and is a finite and continuous function of y in a 
UA 
neighbourhood of the point (a , b) not necessarily including any point on 
the ordinate x — a . 
§ 13. Hitherto we have always had to make some assumption as to the 
existence of ( -£ , although it has been found unnecessary to assume its 
dx * J 
existence at the point itself or on its ordinate. If we make no assumption 
as to the existence of , it is still possible to enunciate a set of sufficient 
dx 
conditions for the equality of all the derivates of derivates at the point 
(a , b), which is a direct generalisation of the Schwarz-Dini conditions. 
To do this we require a theorem of the bounds analogous to Theorem 4, 
and to prove this we require a theorem analogous to Theorem 8. We 
begin with some preliminary remarks and inequalities. 
We had the identities 
m, 
It 
m(a , b ; a + h, b + l c) = , b + k)-m a (b , b + k) 
(1) 
( 2 ) 
* See a paper by the author “ On Non-differentiable Functions,” Mess, of Math., 
September 1908. 
