74 ROYAL SOCIETY OF CANADA 
The quantities, p11, Pon + + + + + Py are in ascending order of 
magnitude between v, and v,. Hence from (12) 
Poni > T; > Ponsa 
It follows that 71 < p,,:, and therefore, 
Pons Pe 7, > ™ 
and so on. Putting a = 41m (ce), we have therefore 
74 Ja 4 LA 
TN RME En = PE EE 
130 = 7,2 FE Oe aes 
a LU Pons Peu 
1/ 
VE 73 7 % 
12 TRE) Pa EUR 
Adding and transposing p,,,,,, we have, 
D2 De Be pe Re NON Une 
Deeds 
Hence by (11), 1 < I*. 
(i)... Properties of C. 
Since p,, P, . . . . . are in descending order of magnitude ex- 
cept at the (finite number of) points v,, v, (see (6), where some of them 
become equal, we have from (3) that p > ” except at a finite number of 
points, ©, v. Since p,, p, eu. 0: Mllare of Class DE Ci" te pe) res 
of class D. Hence the set of reduced curves is inciuded in the totality of 
curves, C, with the following properties: 
ORB aes 
where (1) f (u) is of class D’ for ua < u <1; 
(2) Cts) =O CD) ies; 
(3) f” (u) > 0, except at a finite number of points. 
This also includes those members of the totality, £, for which w’ (t) = 0; 
ie, which have no points of regression, (see 17 —— V”, p.14). Then 
if © furnishes a maximum for the set, C, it certainly furnishes one for 
the totality, £. 
