78 ROYAL SOCIETY OF CANADA 
Hence for v’ other than that given by putting the right side of 
ds Re NE 
(3920; a > 0. Now the slope, v, given by this. is exactly 
the slope of the extremal through 2 by (1). Hence C at all 
interior points has a tangent in common with the extremal through that 
‘ 
ONE oe 
point 7 is positive, and therefore from (4) A 1 > 0. Since 
dv a JE Us, 
du N2u 
has only one solution passing through (1, @)', viz. fj, we have that 
Al>0. Hence again © actually furnishes a minimum, 
S 11. Conclusion. 
The extremal, €, v = & u°, therefore furnishes a maximum. Trans- 
lating into the (x, y)—plane, we have 
dv 
du 
where r and @ are the polar o co-ordinates. The extremal therefore has 
= r,u = cos 6, 

- 3 | 5 &@ 
for polar equation, r= al EL ICON HE 

: . ‘ 3 | 5 @ 
or in Cartesians, x? + y? = ei 
It is to be noted furthermore that these sufficiency proofs establish the 
fact that not only does f, furnish a maximum for curves in some vicinity? 
off, but for all curves of the set, £”, into which the admissible curves 
5@ 
ot 
a! 
in the (x, y)—plane transform. Hence a + y? = | x furnishes 
a maximum for the totality of curves in the (x, y)--plane. 
CHAPTER III. 
AUXILIARY THEOREMS OF ANALYSIS SITUS. 
S1. (a). In § 7 of the preceding chapter, we have used certain 
results dependent upon the division? of the plane into two continua by 


1 Picard, Traité d’analyse, II, pp. 314-5. 
2 See Bolza, l.c., 2 19. 
* See Jordan, Cours d’analyse, 2nd edn., vol. 1, {96-103 ; Schônfliess Gétt. 
Nachr. Math. Phys., Kl. 1896, p. 79; Veblen, Trans. Amer. Math. Soc., vol. 6, 
No. 1, Jan., 1905, p. 83; Bliss, Bull. Amer. Math. Soc., ser. 2, vol. 10, p. 398, and 
vol. 12, p. 336; Osgood, Funktionentheorie, p. 130; Ames, Thesis, 1905. 
