[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 55 
x 
Hence as 7 : 7  — and Y 
— 09 ip fa 
lows that the progressive derivativeesists at 7,, and is equal tor’ (7, + 0).’ 
approach po-itive limits, < 1. Hence it fol- 
Similarly at 7. Hence r (7) is of class D’ on (7, 7,.) 
(ce). The function, U (Tr) 
Rhentanetion, C7 i(z), 16° not defined) for ras) (7) = 0 and 
r (Ts) = 0. But we have just seen that = exists and < 1. We 
" /T +0 
define U (7) as this limit, U,. With this agreement, U (7) is determi- 
nate and continuous on (7, 7,). By (IV: a, b) 
DENT A OT RE NE RES RE CIEL) 
ann =D) Url. Since 7 (r) = 0tor 7, T7 < 7, the: de- 
rivative U”, exists and has the value, 
y (ya’ — ay’ 
Gia UE eI NO AS a eu CI) 
re 
for 7 T < 7,, understanding by U’ the progressive (regressive) deri- 
7) 1! 5 5 8 
vatives at the points, 0. As 7 = 7,, U’ (rt, + 0) is in general indeter- 
minate. From II, U’ = 0 at a finite number of points, r = À, and= 0 
on a finite number of segments, «, <r € A, 2 = 1, 2, 3. 
== 
(d). The function V (Tr). 
According to (I: a), y (yx’ — ay’) is integrable? and has at every 
point of (7, 7,) finite and determinate right and left hand derivatives, 
Hence the function, V (7), is of class D’ on (7, T,) and 
LA) TO Vt) ear eaate oa arleeenterienest esos ax (16) 
with an agreement as to the points of discontinuity similar to that in (b). 
From (15) and (16), it follows that 
3 
Vee 

y 
Dr 
La — = a) a TT ae see (17) 
Ast — 7, + 0, and 7 = 7, — 0, the right-hand side of (17), and theres 
fore also the left-hand side, approaches finite determinate limits.” We 
have from V that V (7,) = @. Further as for U'in (c), V = 0 at a 
finite number of points, and = 0 on a finite number of seginents. 

1See Dini, Grundlagen, etc., 268. 
Fes. Wire We Geil Cay 
3 Dini, l.c., 2191 (2). 
