40 ROYAL SOCIETY OF CANADA 
by an ordinate, is taken as the independent variable. If a denote the 
total area, the integral, I, which is to be maximised, takes the form : 
yout T1 Er] 2 = ¢ EAN en Lee yo 
M 14 i Vdx? + dy , Vdy? + “ 
in Euler snotation. The isoperimetric condition 


HE Versi 000 dx = f"du =a, 
x, 
0 A p >, becomes an identical equation. The 
Fig) : problem is thus reduced to that 
of finding a minimum for [J ay? + du? without any isoperimetric con- 
dition. : ye 
(b). We propose to discuss briefly the transformation under more 
explicit conditions, and see under what circumstances it can be applied. 
We assume that our “admissible curves”! are the totality of curves re- 
presentable in the form: L£:y =f (x), for x, < x < x, and satisfying 
the following conditions : 
(a) f (x) is of class C’? on the interval, (x, . 2,); 
(f) the end-point, B (x,, y,), is fixed, and the end-point, M (x, y,). 
1) is fixed, or 
ii) is free to move on the curve, @ (x, y) = 0, or 
lil) is free to move in any way; 
(7) the curve lies above O x ; i.e., y > 0 ..... SR NL NE RER (1) 
(6) K= vis Y 2 TP =O TOU i = of AC ARE eee ee (2) 
We have to find among these curves, £, one which renders 
I = f°/1+ (4 RARE MIS rr (3) 
To 
a maximum. 

1 See Bolza, Variations, p. 9. 
2 We say that f (x) is of class C’, { ( OP O Aaa any a O10 \ in (x, x ) if f(x) is con- 
tinuous, and f(x) As (DL) NT (x) exist and are continuous; it is of class 
D' | ars if if (v5 X1) can be divided into a finite number of intervals in which 
f (x) is of class C’, ie [Od Ratan) Saute: } ; cf. Bolza, Variations, p. 7. 
