[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 48 
along £'’. In other words, we have a new isoperimetric condition on £, 
and nothing is gained by the process. In case (f” : iii), the curve, 
£", joining (x, yo, 0) and (x, + aN ce y, a) satisfies (B: iii) and is 
an £, since (x,, y,) is perfectly arbitrary in position. Solving (10) for u 
as a function of x, we have; u (x) = y. Hence J’ transforms into J, and 
£’ to an £ joining (x, yp )to(æ + f" i y). In case (f : iii), the 
problems of finding a maximum of I for the totality of curves, £, and of I’ 
for the totality, £'', are equivalent. 
(f). In case (f"' : ii), the condition that the end-point shall lie upon 
p (x, y) = 0,18 
a du 
a yb 0. 
ofatf y V} 
This is a still more complicated isoperimetiic condition if @m (x, y) con- 
tains « If not, i.e., 1f y is constant, then (’ : ii) is satisfied and the 
transformed £” is an £. Transforming into the (x, y) plane by (10), 
the problems of finding a maximum for the totality, £, and of I’’ for the 
totality, £"’, are equivalent if Mis free to move on a line parallel to Ox. 
(g). Excluding from the totality, £, the cases in which we have 
seen that equivalence necessitates a new isoperimetric condition on £”, 
we have the problems of finding a maximum for J along a totality, £, 
y = f(x), where: (a,) f (x) is of class C’ on (x, x,) ; 
(Bo) the end-point, B (ao yo), is fixed and the end- 
point, A (x, y,)° 
i) is free to move on a line parallel to 9x, or 
il) is free to move in any way; 
(yo) y > 0; 
(60) K= [*y de = a, for y = f (x); 
and of finding a maximum for J” along a totality, £’, y = y (u), where 
(a@o'") y (u) is of class C’ in (0, a); 
(Bo'') the end-point, B” (0, yo) is fixed, and the end-point, M’’ (a, y). 
i ) is fixed, or 
ii ) is free to move on a line parallel to 0 x, or 
ill) is free to move along u = a; 
Ge) y > 0; 
are equivalent problems. 
(h). Determination of Constants. 
Euler finds for the minimizing curve in the (x, y)—plane a circle 
through B, centre in AP, (fig.1). In case (fo : i), the other end-point 
