44 ROYAL SOCIETY OF CANADA 
lies in y = y,, and is determined by the isoperimetric condition. In case 
(Bo : ii) we have from the condition of transversality in the (u, y)—plane! 
F = yy’ = 0. 
dy , du 
Since y >0,we have y —0. Hence in the (x,y) plane al = UNE 0. 
The tangent to the circle at B is then parallel to 0 x. This, with the 
isoperimetric condition, fixes the circle. 
(i). Hnd-point, B, not fixed. 
If the end-point, B. is free to move on the curves, 9 (x, y) = 0 or 
to move arbitrarily, the isoperimetric and non-isoperimetric problems 
will be equivalent in the two cases above given. For, for each particular 
B, there is a 1 : 1 correspondence of the type described in (e), and there- 
fore the same is true of the totality of points, B. It might appear at first 
sight that if (xo yo) were free to move on # (x, y) = 0, the con 
dition (6 : ii) need not be modified to (fHo:1i). But from (10), since 
T, — Lo = ee along £’’, the difference is arbitrary. Hence at least 
| 
a 
one of the end-points, B or M, must be free to move parallel to 0 x. 
S2. Sector of Shortest Arc. 
Under similarly indefinite 
M conditions, Euler solves the 
problem: ‘‘Hductis ex puncto, 
O, radiis CM, CA; intra eos 
describere curvam, AM, quae 
‘pro dato spatio, ACM, habeat 
i A arcum, CM, brevissimum,’” 
(Fig. 2) - Here agaiu the point, A, may 
be fixed, free to move on any 
curve, or to move arbitrarily. Transforming as in §1, we find it neces- 
sary to suppose, in order that the problems may be equivalent, that at 
least one end-point, M, is free to move on a circle about C as centre, or 
to move arbitrarily. The solution is again a circle through C and À, the 
third condition being determined as in $1, (g). 
§ 3. General Non-parameter Case. 
(a). Turning to the general non-parameter case, we are required to 
T1 di 
minimize: [= ['" F es y, a eh. HI A NN TC) 
TO 

1 Bolza, Variations, p. 36. 
? Euler, l.c., pp. 144-5. 
