76 ROYAL SOCIETY OF CANADA 
for any u, Ê < u < 1. If now we write 
AI A eh Dee | 
at=f À en du Re SACS (6) 
then from (2), Minis Ag ZAI) fede ash ccnect aeleteom sede nee (7) 
1 ! d 1 1 12 
andéront (ye eg EN NEO TERRE RÉUNIE 
" ne i (o” + 6 7) 

é Paie 
—— pe 1 ud? eT 
_ (5 &\# / 7 du + M. (8) 
3 2 Ô 
from the equation of ©. 
1 

NE 7 (0) — OU Cole Seren (9) 

Since lim. A I; = A I by (7), and lim. 7 (6) = 0, the integral, 
O35 © yes 
0 
12 
ie Be ae Le 
A CURE 0 7')3 
exists and we have : 
ACTE LE URBAN Or ee CEE (10) 
S10. The Weierstrass’ Sufficiency Proof.’ 

U1 (Fig.i4) 
Since by (3), v + 4 7 > 0 within (0, 1), and u>0 for 6, the integral 
on the right hand of equation (10) must be positive unless 7 = 0 within 
(0... . 1) Le. since 7 (t,) = 0, unless 7 = 0), or C coincides with €. 
Hence © actually does furnish a maximum, 

1 Bolza, Variations, p. 74. 
