1908-9.] Reversibility of Order of Partial Differentiation. 145 
The assumptions (2) and (3) are independent, and precisely of the 
same importance in the Schwarz-Dini conditions. Both could be 
omitted if we suitably enlarged our definition of a repeated differential 
coefficient, for without them we could still prove that the double 
incrementary ratio had a unique limit.* 
PART II. 
§ 5. We now proceed to entirely drop the assumption as to the existence 
* df 
of a differential coefficient with respect to y of , and begin by proving 
(JbX 
certain properties of the derivates of ^ with respect to y. 
The following lemma is an immediate consequence of the definitions 
and Lemma 1. 
Lemma 3. — If f x exists at a r point (a , b) and on its ordinate in the 
neighbourhood, any derivate of f x with respect to y is a double limit of 
m(a,b; a + h, b + k). 
For, putting 
mJJ) , b + k) = /( a » & + -/(•*•. ft) 
m(a , b ; a + h, b + k ) = b + k)-m a (b, b + k) ^ 
h 
Therefore, by the definition of a differential coefficient, 
L£ m(a , b ; a + h, b + k) = ^—mib , b + k) at x — a 
h =0 dx 
fx( a , b + k) -f,(a , b) 
k 
Hence, by the definition of the derivates, 
L It Id m(a , b ; a + h, b + k) = the derivates of f x with respect to y at the point (a , b). 
Jc = 0 7i=0 
Thus these derivates are repeated limits, and therefore, by Lemma 1, 
double limits of m(a , b; a + h , b + k). Q.E.D. 
Lemma 4. — If f x exists at every point of an open rectangle 
(a , b ; a + h , b + k), excluding the bounding ordinates, and is a finite 
continuous function of y at every interned point, then, if f(x , y) still 
exists and is a continuous function of y on the boundary, 
L L m(a , b ; a + h , b + k) L U, 
* It should be noticed that the difference between Hobson’s conditions, loc. cit., and those 
given by Schwarz consists in the omission of assumption (3), and that this is only possible 
in the light of the extended definitions suggested by Hobson. If such a definition be 
adopted, there is, it would appear, no reason for making the assumption (2) either. 
VOL. XXIX. 
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