{ WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 73 
(e). Lemma. 
To prove that J’ taken along C exceeds I” along £*, we observe that 
if we have any four not-negative quantities, a, b, c, d,c > a > b, and such 
that a — b =c —d, that at — 8* LB — d*ifO0<a<1. Forconsider 
the integrals 
0 b a a d y C 


A Gel ea Creal 
ET." dz, and af y . dy. They are integrated over equal in- 
0 à 
tervals. At æ in (ab) and y in (ed) such that « — b = y — d, we have 
0 x <y. Hence since a < 1,271 < ya-1. Integrating, 
at — ba < ca — da. 
(f). Proof that I” along C exceeds I” along £*. 
To show that the integral, 7”, along C exceeds J” along £*, we have 
1% =f vA. wu. dt 
tie ithe at RCE se): 
to 
Rearranging as for equation (7) in (d), 
= yas % 4 1 
ef (pe HP RT et pe iu de. (0) 
V1 
~ avo ES 
whereas [= > vy pr SLT OLIS da Pr Pra rr OA Det NE à RC, 010) 
1 02 yy 
From (9) and (10), to show that À — Z*, since the ordinates, u = v 
v 
1) 3 
are finite in number, it is sufficient to show that between v, and v,, 
DES Pe De D OUEN Da an pin) cron ernasenee dns (11) 
(h). We introduce the intermediate quantities, by 
7, e Pon + 1 = Po n1 + Pon, 
eee aa LL 2) 
STEP A ge nt PRE (12) 
Tl Thy-2 Ps Ps 
whence Deen Pi TPs by (3). 
