MATHEMATICS: G. A. BLISS 175 
where^ 
provided that yp is continuous, and of class C" except possibly at a finite 
number of values of x in the interval a ^ x ^b, and provided also 
that the values (x, (p -\- i/, <p' -{- for a ^ x ^ b are all in R. After 
the usual integration by parts of the calculus of variations, and an 
application of the mean value theorem for a definite integral, this becomes 
Ja \ ax/ L aXJx=-x'Ja 
(5) 
where x^ is a suitably selected value in the interval of length h or less 
including x = ^, and on which \p is not identically zero. Let ^ have 
the value 
corresponding to the circular arc in Figure 2, on the interval 
2 2 
where it is not identically zero. 
Fig. 2. 
Then 
(r= J if/ dx = r^ (e- sine cos (9), dh = 2/ sin 0(1- cos d). 
If 6 is allowed to approach zero, while r remains constant, it follows 
that h and 6 both approach zero, and 
im — = \ fy- --fy' + hfy'y'- 
= 0 0- L 
lim 
dx 
5 = 0 6/2 3 L (/a; r Ja:=$ 
(6) 
(7) 
*=o 
