174 
MATHEMATICS: G. A. BLISS 
Further results of interest can be deduced^ provided that the function 
F, the arc (2), and the value ^, have associated with them a constant 
M such that 
<M, (3) 
however 5 > 0, A > 0, and ^ {x) are chosen, provided only that {x) 
is related to 5 and h in the manner described above. 
It is important that these considerations should apply to the integrals 
of the calculus of variations in terms of which the line functions cited 
above by way of illustration, with many others, are expressible. Such 
integrals in general are not continuous, do not possess derivatives, and 
do not satisfy the condition (3), in the forms specified by Vol terra. It 
is the purpose of this note to prove this statement, and to call attention 
to the modifications of Volterra's definitions which apply also to the 
line functions of the calculus of variations. 
Consider the simplest type of integrals of the calculus of variations. 
F[y{x)] = JV(^, y{^). y'{x))dx. (4) 
The length integral is a special case which is not continuous according 
to Volterra's definition. For in the figure ac + ch 
is the length of each serrated line joining a with 
h and consisting of the slanting sides of the tri- 
angles with bases on ah and equal altitudes. There 
is one of these serrated paths in any neighbor- 
hood of the straight line ah. Hence the length 
integral is not continuous according to the defi- Fig. l. 
nition given above. 
A function will be said to be of class C^"^ if it is continuous and has 
continuous derivatives up to and including those of the n-th. order. 
Consider then a curve (2) of class and let the function / in the inte- 
gral (4) also be of class C" in a neighborhood R of the values {x,y,y') 
belonging to (2). Then the increment AF for the integral (4) is expressi- 
ble in the form 
= j"*' {J{x, <p^^p,^' + i') -fix, <p, } dx 
= £{AxP + BxP'}dx, 
