176 
MATHEMATICS: G. A. BLISS 
where the arguments in / and its derivatives are x, (p, <p'. It is clear 
from these expressions that at a value ^ defining a point on the curve 
(2) where the derivative fyy is different from zero, the limits (6) and 
(7) may take any arbitrarily assigned values, one value only being ex- 
cepted in each case, provided that r is properly chosen. The function 
(4) has therefore no derivative in the sense of Volterra at the value 
X = ^, and does not satisfy the condition (3). 
For any more general variation \p satisfying the restrictions specified 
in the first part of the last paragraph, it is clear from (5) that 
a iJo \ dx / J Jo J«=«' 
where the arguments of / and its derivatives are x, (p drp, <p' -f d\l/\ 
Hence if the conditions 
as well as those described above are satisfied, the derivative limit will 
exist and have the value 
the argimients of the derivatives of / being x, (p, <p'. Let N be the maxi- 
mum of the absolute values of fy — dfy> /dx, fyy, fyy, in the neighbor- 
hood R. Then from (8) 
AF 
AF 
<r 
< 
AF 
dh 
0- 
8h 
and it is evident that the quotient AF /bh is boimded for all values of 
b > 0, h > 0, and xf/ such that the inequalities (9) hold and the values 
{x, (p \p, <p' -\- \p') ior a ^x ^ b are in R. 
By similar arguments it will be clear what properties are possessed 
by an integral of the form 
Let the curve (2) be defined by a function (p of class C " . In a neigh- 
borhood R of the values (x, y, y', . . . , y^"^) belonging to the curve, the 
function / is supposed to be of class C^""'"^\ Then^ the function F has 
continuity of order n. In other words, for a given e there always exists 
a b such that 
\^{x)\<b, i^'W|<5, |^''W|<6 
{a^x^b)y (9) 
h = 0 
