72 ROYAL SOCIETY OF CANADA 
Then C also passes through (1, &). For 
RCE ET sin dy 
u-1 V1 V2 Vy 
= yf (D, sc en med A dey rab Ru) du RE EN. (3) 
U2 vy 
Now we have seen that uw is an increasing function of t on the are, 1, 
(see ST, (e) and equaticn, 1), and that it is alternately decreasing and in- 
creasing on the arcs, 1, 2,3 . . . . (2n + 1). Hence « is a decreasing 
function of ¢ on the arcs, 2, 4,6 . . , . 2n, between v, and v,. Now 
by CUI” 'a TN” se); 
D (1) i vi dt=f'p.w. dt daha aa 4 rer RE (4) 
ty to 
We consider the contributions of the arcs of £* from v, to v, to this 
integral. Onthe arcs, 1,3,5 . ... . .. 2n =} 1), simce u (2) is anom 
creasing function of t, if we denote by 7, 7’, + — 7’, the parameters of the 
end-points, then u (Tr) = v,, and u (Tr) = »,, whence 
T1 Lo ’ 
Fi Panini 00e ot = / Pon 1 + UU sense nues sous (5) 
if V1 
expressing as a function of u. On an arc, 2,4, 6 . . . . 2n, from 
r to 7”, r <7, since u (t) is a decreasing function of t, u (T) = v,, and 
, à 
u (D) MEence 
iF eat v2 
JP» AU QUI =f Pepa NE je oe UMP 0e Léon eee CORRE (6) 
T Vo V1 
Rearranging and combining the ares between v, and v, we have: 
v (1) JU | &, Sa Dah aie aw = 0 dette. elie Pon + 1) . au en OT 
From (3), (4), (5), and 111” : a), we have: 
D(A) wr) nc ER CPP ARENA nee see CON 
‘ 
