[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 75 
§9. Sufficiency Proof by Taylor's Theorem. 
(iw) Te v=W.U 

0 U-| 
(Fig.13) u 
In the present case, we can prove that © furnishes a maximum for 
the totality, £*, by the remainder formula in Taylor’s Theorem. As in 
€7, if us < 0, we adjoin to C the line, 0 ws, and denote the resulting 
curve by C*. Since = = 0 in 0 wa, if us > 0, it follows from the defini- 
Lu 
tion, (see p. ) that J” along Cis equal to J” along C*. Now 
Oe ie te CR (1) 
YO 
Ryle eee 
We write À = y + 7. Then ris of class D’ in u, (see §8,1:1). We 
denote by 7/ the derivative of 7 with respect tou, Expressing /” as an 
| 14 14 
integral, VAUT =! | — (vd + my WU. AU cacccseessscese (2 
0 
Now if us > 0, © = 0 on 0 wa, and ÿ — f (u) > 0 from us to 1 
| 5 ; 
($S8,:13). Hence ÿ > 0. Again v => &, u,? and therefore, v’ > 0 for 
u>0. We surround the origin by a small 6 — interval, 0 < 0 < 1. 
Then ford < u < 1, v’ > 0,and y > 0. Hence if 0 < 0 < 1, 


DE MO RNA een terne ee (3) 
If now we write: V = (@! + A)”, we have 
1 2 1 
oe eat 31 RAT PRES 0 or = — - ra 5 soso seneee (4) 
oh A (D PE CRE NOR UE VE); 
2 
From (4), AS and ca exist, and are continuous functions of h 
oh Oh? 
on0 . . . . 7” except perhaps for À = 7’. It follows! that 
7 il 7° 
DH É = DE 44-4 EE —— 
( + ) v a $ ys 9 (v’ a a) n'} 
1Stolz, Diff. u. Integr. - rechnung, s. 97. 
