[WILson ] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 61 
Hence if u’ > 0 02 £, as t = k + 0, and t — ki+ 1 — 0, we have 
OU LE A Ae ; 
u—u,+0,and u + u — 0, while de approaches finite limits, viz., 
(+1 
p Qu, — 0) and p (u;,, —0). It follows that the progressive derivatives 
of v (u) at w, and regressive at u;,1 exist, and are equal to these limits. 
Hence v’ (x) is of class C’ on £,''! If we write 
Du) thentp— Ju) 
(h). The curve, £”, then consists of a finite number of ares, £,", 
such that 
. 
(A). Conditions on Single Arc. 
Leu Tu); 
GR (sof class Chine) 2 = 0102). dr; 
(2) f, () = @; 
(3) f:’(u) > 0 in (u; u; 4 1) except that it may be that f,’’(u,) :: 0; 
py / p” / 
D) me |) te Nr | exist and are finite ,” 
OE gaan HN ra : 
Tes 
{UU 


(B). Conditions for Composition of the Arcs. 
GNU 0<wu <1 except on £,” where, if p (u,) = 0, it may be that 
u. = 1, and on £,.” where always, u, : 1 = 1; 
(2) (a) iff, Cu) = f;" Cu), then? = 7, except possibly atu; , ; where 
eet + 1; 
cb) if f7"(uj41) & fj +1 Cuj + 1), then fi’ (u; 4 1) does not lie 
between them ; 
(c) if (us) > 0, then f (ue) > fo (ue) for i + 0; 
(3). The compound curve is continuous. 
Conversely, if any curve be made up of the finite number of ares 
y, MW , 
v' =f; (u), subject to the conditions just enunciated, it is not difficult to 
show that we may select a suitable parameter, (e.g. the are-length) so 
as to exhibit the curve as a member of the totality. £”. It follows that 
the problems of finding a maximum of I’ taken along a set of arcs, £: 
and the curve, £"”, are identical. 

1 Dini, 1. c., 368. 
2? These follow from (III’ c). 
