46 ROYAL SOCIETY OF CANADA 
for a path in the (a, y)—plane from (as, y,) to (&, y,;) where & = x (a). 
The problems of finding a maximum for J among the curves, £, and 7” 
among the curves, £’’, under suitable continuity conditions, are therefore 
equivalent. 
(c). It is evident that the method will be effective also if H con- 
tains u = ve G (a, y, i) dx; i.e., if the integrand of (17) is of the 
form, H (u, y, y’), and Fis of the Form r} vit G . dx, x, T1 
To 
S4 ÆExample. 
As an example, illustrating the last remark and at the same time 
showing the necessity of definitely formulating the end-point conditions, 
let us take a third problem of Kuler’s: ‘ Jnter omnes curvas isoperi- 
metras, definire eam in qua sit ac s". dx maximum ver minimum, denotanti s 
arcum curvae abscissae x respondentem.” We suppose one end-point fixed 
and take it for origin, the other is free to move ona = x, We assume 
that our totality of admissible curves is the totality, £, 
£:y=f(), 
where: (a) f (x) is of class C” ; 
(8) K=f"\) pat (2) AE al antec (19) 
a 
We have to minimize, J = we CAP CE EE animes ARE AR see mee accent (19) 
0 
Transforming as in $ 1, to s as independent variable by the equation 
he Vit+(Z)-% id ee (20) 
we Obtminy Fhe fe oe gS) aaa cree (21) 
to be minimized for a totality, £”, in the (s, «)—plane: 
I (8). Lor O< sels 
where: (a) x (8) is of class C’’ ; 
CB") O° ae AIRMAN Uipse acti cepeetes ccaseescind ticosttgesns anata eee (22) 
and £’’ joins the end-points, (9, 0) and (4, x,). Making the transforma- 
tion of 7” back into the (x, y) plane by: 
SO Tey ee Bee acres ee (23) 
