1908-9.] Reversibility of Order of Partial Differentiation. 155 
m(a, b; a + h, b + k), they lie between the upper and lower double limits 
of m(a , b ; a + h , b + k) when h and k are indefinitely diminished. 
To prove this let v(x , y) denote any derivate of f(x, y) with respect to 
x , e.g. the right-hand upper derivate ; then, for each value of y there is a 
sequence of values of h which determines the value of v(x , y) as a limit, 
viz. 
Therefore 
v(x,y) = L 
the derivates of v(xy) at (a , b) = hit ht ^ i /O* + h kttli b + k) — f(x, b + k) 
k = 0 n— co tv ( ljc,n 
_ f(a + h t) ; n , b + k) - f(a , b) ) 
K , n ^ 
Changing h k>n in the first fraction into h 0>n , we get m(a ,b; a-\-h 0n ,b-\-k). 
Remembering that the sequence h k<n gave us the highest possible limit 
for the fraction, this change may diminish and cannot increase the value 
of the right-hand side of the preceding identity. Hence 
the derivates of v(x, y) \ hit L t m(a , b ; a + h 0>n , b + k) 
k = 0 U— oo 
A a double limit of m(a , b ; a + h, b + k) 
A lowest double limit of m(a, 6; a + h, b + k). (1) 
Similarly, changing h 0 <n to h 7 — n , we get 
the derivates of v(x , y) A greatest double limit of m(a , b; a + h, b + k), (2) 
which proves the statement made above. 
Hence it follows that, if the limits of m(a, b; a + h, b + k) are finite, 
the first derivates with respect to one variable are continuous functions of 
the other variable. 
In particular, this will be the case al every internal point of a 
rectangle throughout which m(x , y ; x', y') is bounded. 
It must here be pointed out that, in speaking of the derivates of 
derivates, and in deducing the above results, we have tacitly assumed 
that the first derivates are finite, or quasi-finite. Unless this is the case, 
their derivates are not properly defined, and it would only be by intro- 
ducing fresh conventions that we could apply the above results to such 
cases. Thus, for instance, if f(x , y) is the sum of a function of x and a 
function of y, m(x , y ; x r , y') is zero always, the same is therefore true of 
the repeated derivates, where they are defined. If the function of x has 
an infinite derivate, or differential coefficient anywhere, in particular if it 
is a non-differentiable function of x , in which case such points are dense 
