[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 8 7 
III’: Initial Conditions : 
(CON ARS CNET NE CHIENS 
(DIARGIE (Diss OG A ale 
% 
% WA 78 
FU NAT d Pp pe exist, and are finite 
Dr =a eee 29 OL : Nr ata ‘ 
are re Ua Tot 


LV’: Regional Conditions : 
(ane <<) OS ior tor OT 
(CS (a) Onor ER 
The isoperimetric condition becomes : 
da 
K=/V'.dt=V(n)-VG)= 0 
To 
by (III’: a), and is therefore satisfied by all the curves, £’. The integral, 
I, takes the form 
Ty 
% 4 
NC DL et es Pare (21) 
1 
(4 
Our isoperimetric problem is thus reduced tothe non-isoperimetric prob- 
lem of finding a maximum value of J’ among the totality, £’, one end- 
point, (1, @), being fixed, and the other end-point, (U_, 0), being free to 
move on the axis, v = 0. 
(h). Conversely if any curve, 
A = OG ew = Vir) 2222 A fein ve MT (22) 
is given, satisfying the conditions set down in (/), and we define : 
B 4 ee 
Gi) OMR DAT CONNECT | 
it is not difficult to prove that the curve : 
2 = p()y¥= 4%), 
belongs to the set of curves, £, given in (c) and furnishes for the integral, 
ay vY F sun, a 
[= 3 CHERE eh 
the same value as the curve, £’, does for the integral, 
ic 
: 7 ae 
DRE UO. dr 
: 
