8O ROYAL SOCIETY OF CANADA 
two regions into which the auxiliar lines, a a and a b, divide the plane. 
The product, 
G (wy = a q, CODE h, (Cam Dine k, Gay 
for the whole plane then has the essential property of the G-function in 
the article referred to, viz., that in the neighbourhood of the zeros, the 
factors change signs in pairs except at points on the curve itself. 
Proceeding in a fashion entirely similar to that there given, we may 
prove that the totality of points in the plane fall into three classes : ! 
(1) points, (x, y), such that @ (x, y) has both signs in every vicinity 
of (x y), however small; 
(2) points, (x, y), such that G (x, y) takes positive but not negative 
values in every vicinity (sufficiently small) of (x, y) ; 
(3) points, (x, y) such that @ (x, y) takes negative but not positive 
values in every vicinity (sufficiently small) of (x, y). 

(Figie) 
The points of the first class turn out to be identical with the curve, 
©. Using the auxiliary lines, | 
VRAIES at Cue lee ttames 
and joining them up by ares of circles of radius, €, we may construct two 
auxiliary curves’ consisting of points of classes (2) and (3) respectively 
as near as we please to C.2. By means of these we may join any two points 
of class (2), or any two points of class (3) without meeting C, showing 
that there are just two continua. 
$S 2 (a). An ordinate can meet an arc of type, y = f (x), at most once, 
Since there are a finite number of such ares, it follows that any ordinate 
not through a straight line parallel to 0 y must meet the curve at a finite 
number of points only. Consider any ordinate through a point, 1 (£, 7), 
(see fig. 17), interior to an arc, C,, y = f, (x), end-points (a, y,) and % 

1 For details, c.f. Bliss, L.c., vol. 10. 
? See fig. 16, and for details compare, Bliss, I.c., vol. 10. 
>This construction applies equally to any curve consisting of a finite 
number of ares of type, y = f (x), or x = f (y), including curves of class D’. (See 
Ames, ‘lhesis, already referred to.) 

