[ WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 45 
for a set of curves, £, y = f (x), joining (x, ye) to (x, y,), satisfying 
certain continuity conditions, and the isoperimetric condition, 
DS LL x f dy om 9 
= Gay. de =1 sn c AE AREAS SRE (12) 
to 
We suppose that at least one end-point, (x,, y,) is free to move in a 
straight line, y = y,, parallel to O x. The other end-point, (2, y,), may 
be fixed, free to move on a curve, or unrestricted in position. But a 
minimum in the last two cases will also be 4 minimum for the subset of 
curves passing through the end-points of the minimizing curve. Hence 
we find the extreme in all cases if we find it for (x,, y,) fixed. We sup- 
pose that G (x, y, A) > 0 for every finite x, y.! 
(b). Transformation from the (x, y) to the (u, y) plane. 
As in § 1, we introduce the new variable, u by 
u= f° {1%}. az AAC PEN UE LCL CARPE (13) 
ro 
Since G > 0, we may solve this for x as a function of u, subject to cer- 
tain continuity conditions. Substituting in y = f (x), we obtain y as a 
function of u = y (u), satisfying certain continuity conditions and 
tog Gi, pele 0, A Fk ER RR AA METI el (14) 
E 
where x’ = =. etc. The integral, Z, transforms into 
du 
- Aie y ! 
VE =i) TX . HIT, Y, ae RC i ea AP CaP APES e (15) 
0 
aiong a path, £’’, in the (u, x, y)—space, satisfying (15) and certain con- 
tinuity conditions. If we can eliminate x from (15) by means of (14) 
Pare = f TGS DNS ARE SAN ER PE re (16) 
along a path, £’, in the (u, y)—plane. Conversely, if we can solve (15) 
for x’ as a function of x, y, y’, and therefore of x and u, and if the con- 
tinuity conditions are such as to permit, x' = f (x, u)* to be integrated, 
we can obtain x ‘u) satisfying (14) and such that x, = x (0). Since 
G > 0, we bave x’ (u) > 0. We can therefore solve x = x (u) for u as 
a fnnction ofx. Substituting in /’’, we obtain 


1If G(x, y) <0, we change the sign of Gand l. 
* See Picard, Traité d'analyse, II, chap. 11. 
