64 ROYAL SOCIETY OF CANADA 
acand a; +4, of £;'’, it follows from the continuity of £.’’ that this rela- 
tion holds up to and including these end points. 
(f). It remains to show that there are no corners.’ For this pur: 
pose we select points, P, (7) and P; , , (r”) within the arcs, £:’, and 
Oe, and let a (7)'= a ur") =a" Consider the variation, 
v=f (4) + em (hu = a, 

where : (1) 7, (u) is of class C’ in (a’ a; 4 1), and (a; 4 1 a”), 
(2) mi (@) = 0, m (a) = 0,m" (a) = 0, 
man (0) = 0, m5 410") = 0, me à (0") = 0, 
mas La) = 0/7 (a; La) = 0,7 Gee 
Wi a (Gi sa = 0, 72 2 GREEN 
(a +1) = 0; 
(Dr @ = or y = Gel 7 (DEN 
(a a’), and 7 (u) = 0 for win (@”’ a; , 3) 
Since such a variation is made only at the corners common to two 
successive arcs, condition A (4) is not affected thereby. It follows 
therefore exactly as in §6 (b) ..... (e) that for |e | < §, this variation 
is amissible, whence from the result in the general case. 
C'; a 1 = Cy’. 
There are therefore no corners. 

1See Bolza, Variations, p. 68. 
> ni (u) = (u = a’)* (U = a, 4.4), and 7; , 1 (u) = (u- a!) OI - a; , ;)® satisfy 
these conditions. 
3 See Bolza, Variations, p. 38. 
