[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 4] 
(c). We now introduce, after Euler, the variable, u, where 
taken along £ . From (4) and (1), 
du 
dy = 0, 
whence we can solve (4) with respect to x,!and the inverse function, 
Baa EU) RENE ER PRE (5) 
will be of class C’’ in the interval (0, a), and it will increase from x, to x, 
as u increases from 0 to 4. Further, if we define, y (u) by 
yu) =F {acu} 
then y (u) will be of class C’, on the interval, (0, a), and between x (u) 
and y (u) bolds the relations : 
DATA NT EE OC RO EC RER TETE (6) 
By the introduction of u, the curve, £, in the (x, y)—plane is trans- 
formed into a curve, £’, in the (x, y, u)— space: 
\ L':æ = x(u), y = y (u), 
where. 
(a’) x (u) is of class UC” and y (u) is of class C’ in (0 a); 
CB) x (0) = ay y (0) = y and x(a) = xy y (a) = yy where 
EL, Yo a) 15 
i) fixed, or 
ii) p (x, y) = 0, or 
iii) x, and y, are arbitrary ; 
(y) y @) > in 0a); 
(6) y x' =1. 
The isoperimetric condition is expressed by (6’). For, if we introduce 
u into Æ by means of (0’), we have 
Fe ge ER du of ak 

' Osgood Funktionentheorie, p. 193. 
