62 ROYAL SOCIETY OF CANADA 
$6. The Extremal. 
(a). Let us suppose that some curve &, of the set, £’’, (fig. 6), 
furnishes a maximum for J’, and that (© is divided into arcs, 
Lo yn ie MEME Of type, ef.) as im the preceding pana. 
: ARE : une 
y | graph. Let eu ne any two potas in 
a 7 the interior of 2i\ and) WG 
! ag u (7") = a”. Choose a variation, 
ete) oh L @ } 
Len Lu (t) t where 
1) 7) (u) is of class C’ in (a’ a”); 
1) Ca) 057 a) 0g ee 
ma) = 0 9) Ca) =O a ie 
Gal: LD) EU outside of (7' T2) hen 
ttl the curve, £” (u, v), is an admissible 
(Fig.6) variation ; 7.e., belongs to the totality, 
£'', described in § 4 (c). To show this, since £."”' is the only arc affected 
by the variation, we need only show that (non-zero) limits of |€; can be 
flxed +o small as to satisfy A (3), and B (2), of the preceding paragraph. 
(b). To obtain these limits, we observe that since 7’ (t) is con- 
tinuous, there is an upper limit, m’, for ||". Since (7’7’’) lies within 
x", v + 0 and u’ + 0 o0n (77). Since these are continuous, there is a 
positive, v,u<|u'|, and v < |v'|, on this interval. Further, if there are 
any points ¢’, on £,;’, 7 + 7, such that u ({) = u (t) for r <7 <7, then 
the difference | f;" (uw) — f;" (wv) is a continuous function of u on (a’ a’’) 
and + 0. Hence there exists a 9 such that 


dia a! 
lA @ —f Wis pS 0 
If now & be any positive quantity, 
then for any ¢,0< t| < &, £” belongs to the set of admissible curves. 
(e)2.) Simee. (a) Wiel <a577 << dl 
m 
Co) ae ese pow 
(Cnil 

! Osgood Funktionentheorie, 213. 
* Dini Grundlagen, etc., s 68, 70. 
