[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM FFG 
: 1v : : ~ 
Since — > 0 except at a finite number of points, ($8, i: 3), v > 0 
except at ui. There is therefore one and but one extremal joining 0 to 
a point, 2, on C,2 = uw, viz: 
D = EN Sescles SBE ACOB a le tie ces sa en UNE RC Il 
wig (1) 
With the usual notation, we write, 
U2 ‘ es ft : 
S (u,) = | HS A ENT LEE NO 40 AST ARS Bot 
(2) Ug 
for wa <u, <a 1 Then S§ (1)\.= Wa along ©. Again, as u, = u,, 
0 
S (u,) = 15 along C. We define S (v,) as this limit. Then S (u,) is 
u-1 
continuous On (u1 . . . . 1). Again, except for u, = wn, 
as 2 74 5 1/ of 1 2/ x: 5 ae 5 
dus CG) To ea ea a EN Mo UR Ci Ux 2a te) 
2 Œ i 
As u, = uw, this approaches a definite finite limit, viz, — V1 . wa, 

ds : = : CT OL 
Hence —— exists progressively, and is equal to this limit.” Then 
SL 
AT = f ys Oho Oli We v' 1, u du 
= 4 a a . . ‘3 




0 u-1 
= SL) S (0), 
1 dS 
= f du, Ms du; ns nn eee ee sons ee esteccsece (4) 
u-1 x 
. 5 LS 2 74 5 14 24 
Now from (3), if v’, = 0, Sep e uae SVU) A tbs, When 
du; 3 = 
ads Sas NU : À 
i > Owithin C. If v’, = 0, consider the values of the right-hand of 
Us 
(3) as a function of v’,, It has a maximum or minimum when 
PES ae 
(2) 2 Fe 4 2 Us Se COS erst eee nouer Fe HEHEOE SEH O ETE EE (5) 
3 48 Be aE : are : 
the latter since ee Tah a < 0. This minimum value is 0. 
Tie & 

» Bolza, Variations, p. 87. 
2 Dini, l.c., $68. 
