66 ROYAL SOCIETY OF CANADA 
r, and for 7, when it is not already determinate we define @ (r,) and 
0 (7,) as the limits thus approached. Then the are (7, 6), joining 
iP. \7 ADN) EAU | ,and Q 7 = SC Gr) tis continuous (see 
III: a). If6(*,) > 0, the curve made up of this arc from P to Q, the 
axis, 7 = 0, from Q to 0, and the axis, 6 = 0, from 0 to P, (fig 9) is closed. 
Et is’ also simple. For ‘since’ 2b o>) (EW eb): torino) <\G =i ee 
6 = cot” . Dir a we have @)(7)) = 0) for75 << ae (2) 
: wp a 
Hence P Q cannot meet the axis, O P, (0 = 0), for any 7,7), <1r< 1. 
By hypothesis at Q (r.), @(7o) > 0. Hence P Q meets O Pat PG) 
only. Again by equations (1). since + (r) > 0 fort, <1 <7, (IV: b), 
while 9 (r,) > 0, we haver (T7) > 0 for 7, <7 <7, Hente that are, 

(Fig 9) (Fig.10) 
P Q, meets the axis, O Q, r = 0, at Q only. Further, the arc, P Q, can 
have no double points on its interior. For if there were such a point, 
T=) TT. we have from (1), 

aly (73 
PC) + CD = PF C)+HG,), and 29 _ FC) 
2 3 up Ge) yp (7) 
whence @ (7,) = @ (r,), and # (72) =  (t,), contrary to (I: b). 
Similarly if 6 (r,) = 0, we may show that the arc, PQ, and the axis, 
4 = 0 from Q to P form a simple closed curve. 
(c). Since p < 0 on the interior of £ (LV : b), we have 

Pea NP Gere PRE 
i 2 
Hence #'= 0 on the segments, «; A; and vanishes at only a finite num- 
i, 
ber of other points, (IL: a, b). Onthe former, 6 = const. Entirely after 
the manner of § 5, we may ae the remaining ares as a finite number of 
continuous arcs of the typer =f (@). It follows that if any ordinate be 
