[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 69 
them and v for which p; (u) — pj (u) <6. From (5), these have v” for 
an upper bound, and hence have an upper limit, v’’.' By (5), 
v<v' <v". Since v’ is a limit-point for a set, [u], for which ; p (u) 
— pj (u) < 0, we have, remembering that p (u) is class D, 
pov = 0) — pv" — 0) > 0. 
rom ep. 13); p, (@” — 0). == p (0°, — 0). Hence 
1 (OL ON ERIC) 2 OK | meray een (7) 
On the other hand, if p; (v’ + 0) — p; ("+ 0) 0, since p; and p; are 
of class C in some vicinity of v” for u > v”, p, (u) — p; (u) < 0 in some 
vicinity of v” for u > v”. Hence v” is not an upper limit for the set for 
which p; (u) — p; (u) <0. ByIl’:c,p, (v” + 0) = pj (v" + 0). Hence 
p; (v" +0) — BD, (CY F0) HO 0... cee 71. (8) 
Now the intervals, | Pi AA aU SE IN MA co Pear aise ent a and 
1Pi (os |) eer eat kee ar Py Or 0) | cannot have any point in 
common. For if they did, either p, (v” + 0), or p; (v” — 0) must lie on 
{23 CO) TEE Per) —00) \. contrary to (II”: c). 
Crary) neat ae) CE 0 up (Bt tO) 
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Bigs oT. 
The inequalities, (7) and (8), are evidently incompatible with this con 
dition (see fig. 11). Hence the order of magnitude of p,, p; Pa ... . is 
independent of v when u = v passes through no discontinuity of p. 
(g). From this it follows that at any point, k, of discontinuity of 
pj Or pj, the slopes, p, 2 Py» + . . . are in the same order of inequality, 
where by p; (x) either of the values, p, (k + 0) or p; (k — 0), is meant, 
and similarly for p; (k). To see this, we select any v’ and v”, vo’ < k < v” 
such p; (u) and pj (u) are continuous on (#’, k) and (k, v”). Then, as in 
(b), we derive p; (k — 0) > pj (k — 0), and p; (k +0) <p; (K+0). (9) 
Since the intervals, p; (k- 0)... p; (k + 0) and p; (k— 0) . . p 
(k + 0), can have no point in common, it follows from (9) that p; (k—0) 
and p; (k + 9) are both greater than p; (k - 0) and pj (& + 0). Hence 
the inequalities, (4), hold for every v,v, <v <v, Further, since p (u) 
is continuous on each £,” (see $5), at the points v,. v,, we have 
JO aS OE ay ree ree oe Serine ae 8" (10) 

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