48 ROYAL SOCIETY OF CANADA 
and the continuity conditions permit of the integration of this equation, 
we may obtain an æ (t) satisfying (: 6), and such that x (f,) = ao Sub- 
stituting for u’ in I” from (25), we obtain the integral, J, along a path, £, 
from (2, y,) to | EAU) AU L Under suitable continuity conditions the 
isoperimetric problems in the (x, y) plane, and the non-isoperimetric 
problem in the (u, y) plane are equivalent. 
S6. General Remarks. 
(a). Inthe first example of Euler’s which we have given, (see $S 1, 2) 
the set of admissible curves in the (u, y) plane contains the whole set of 
curves of class C’ in a certain region containing the end-points. Varia- 
tions of the type used in the proof of the fundamental lemma on which 
Euler’s differential equation depends can therefore be constructed.! In 
the third problem, however, the transformed curves are subject to the 
slope condition 0 <|a'(s)|< 1. The question therefore arises 
whether it is possible to construct a variation of the required type with- 
out violating this condition. Indeed, if we use parameter ‘representation 
and admit a corner in this problem we find in the (u, y)—plane that an 
admissible curve in the (s, «)—plane satisfies the conditions, 
AO ate == tas IS) Non (0): 
where : (a) x (s) is of class D’; 
(pa) Oye WS)iieons 
(A) « (0) = 0,2 @) = x. 
| 
\ 
\ 
à | 1 
ie ! I 
Di CHE 
NP | 
| 
| |. MS 
Te Cr 
The broken line, O0 AP, constructed as in fig. 3 so that | #’| = 1 along 
it is therefore an £’’.. But in order to obtain such a solution, the method 
employed is to vary OA, keeping À P fixed and vice versa? It is evidently 
impossible to construct such a variation without violating the condition 
[ee NU: 

1See Bolza, Variations, §5. 
2 See Bolza, 1. c., §9. 
