[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 79 
a particular class of simple closed curve. Using Cartesian co-ordinates, 
our simple closed curve, €, given in the form, 
See (t), ie p (t), 
can be divided into a finite number of arcs. 
(a) of type y =/ (x), f denoting a continuous function, and 
(b) straight lines parallel to 0 y. 
The method used by Bliss in his first article (see below) for proving 
the section of the plane into two continua by a simple closed s curve 
consisting of a finite number of arcs of type (a) may be readily extended 
to the present case. Through the end of each arc of type (a), (the are, 
C, of fig. 15), we draw half-lines parallel to 0 x. These with (, divide 
the plane into two regions. As in the article referred to, we construct a 
continuous function, g, (2, y), which vanishes on these lines and these 
lines only, and takes different signs at points (x, y), in these different 
regions. Ifa line, D, of type, (0), have for equation, 
HENTAI 
the function, h,(x, y) = x —x, 
a 

has the same properties with reference to this line. At the intersection 
of an arc of type (a) with an arc of type (b) (see fig. 15) at (a), we 
construct the function, 
ky (@, y = (@ — 2) J 2 forz —a>y — y, 
(i=) / 2 for æ — M <Y — Yp 
supposing that the auxiliary half-lines already drawn from this corner 
‘are in the positive directions with reference to 0 x, 0 y as in the figure; 
a similar function may be constructed by appropriate change of sign if, 
in other directions. This function is continuous, vanishes ouly at the 
broken lines of the figure, and takes opposite signs at the points in the 
