[WILSON] CERTAIN TYPE OFNISOPERIMETRIC PROBLEM 47 
it is not difficult to show that the problem of minimizing J for the totality, 
£, and J” for the totality, £’’, are equivalent. Applying the usual method 
to the latter problem, we obtain the solution,’ 
$ = const. 
This is not an admissible solution, since in £, , > 1. There is there- 
ax 
fore no solution under the initial conditions specified. With unspecified 
conditions, Euler obtains a solution.’ 
$5. General Parameter Case. 
(a). From the preceding, it might seem that the introduction of u 
as independent variable was equally essential with the possibility of 
eliminating « or y in the transformed integral; and therefore that the 
condition, G > 0or G < 0, is indispensable to the success of the method. 
But when we permit ourselves to use parameter representation, this re- 
striction may be removed. Denoting now by 2’, y’, derivatives with 
respect to the curve parameter, {, we suppose given totality of admissible 
curves to be: 
Lic= p(t) y= (d), 
joining 0 (t = t,), and 1 (¢ = ¢,), satisfying certain continuity con- 
ti 
ditions and ni ve Cio Gea Le GED NAM Feel PRÉPA sonnei (24) 
tO 
We assume again that one end-point, 0 (x,, y ), is fixed and that the 
other, 1, is free to move on y = y. Weintroduce the new variable, u by 
ACER Cen roy a OM ae pone tenho (25) 
to 
or Ta Ui) eG HEE OU) a en EE ER nee ele Des e (26) 
The method is effective if x, (or y) can be eliminated from (26) and the 
t 
inte: ral, = he 'F (NON à WN be wen cseweccs nan atone tnlnces ccuasen ss (27) 
to ‘ 
which is to be minimized. We obtain, 
11 4 
DE Cae ee aCe tir Eee a on eee (28) 
t? 
for a curve, £”, joining (0, y,) and (J, y,), and satisfying certain con- 
tinuity conditions. Conversely, if the equation, (26), can be solved for 
az’ as a function of a, y, y', and therefore of x and f, 
Eh fe (a, i); 

1Bolza, Variations, p. 20. 
* Euler, l.c., p. 208. 
