[ WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 63 
it follows that 5 = v’ + e 7’ (t), and v’ have the same sign, and do not 
-/ , 
2 / . 7: U A > Vv c . 5 . 
vanish on (r’7'’). Since fi (u) = =, and 4’ (u) = —;,' (3) is satis- 
é u u 
pv os we oe 
7 1: > 7? > > 11 
fied. Since < 6 < PG we have | fs — fi = a HO AS) TENTE 
for j + 7, and | fj’ — fl | >”, we have | fj —fi 1 > 0, whence B:2 (a) is 
satistied on the interior of Lv’. 
(d). Let us suppose, if possible, that B : 2 (b) is violated for 
some €, 0 < le,| < & and some particular w. From the equation, 
/ [41 
Fu eA +s 1) Fa 
Hore — 0, re. for f,,\f, does not lie between f”, 1 (uz + 0) and 
f!" (uy, + 0, while for e = e,; it does. Hence for some 
nf ne (ux + 0) J () fe (uz + 0) 
| | 
(Fig. 7). 
is certainly à continuous function of e for u = ux. 

| 
intermediate value of €, f,’ Ce) = fj’ we + :0), or fi (ux + 0. 
This, however, contradicts what we have proved in §6 (¢). Similarly for 
B:2(c). We have therefore finally that the variations of SG (a) are 
admissible variations for every 0 <|€| <& 
(e). It follows in the usual way * that £;”" from 7’ to r”' satisfies 
the equation, 
=a Uh = const. 
Since wu’ + 0 on (Tr), we may solve for { as a function of u, and sub- 
stitute obtaining the relation 
dv 
=> = CU 
du ; 
; 5 
whence VSO yp Ui + C'ipu. 
Since -’ and -’’ may be selected as near as we please to the end-points, 


1 See $5 (a). 
2 See §6 (b). 
3 See Bolza, Variations, p. 22. 
