84 ROYAL SOCIETY OF CANADA 
C, and C, are of the same species, and by the senses of description are 
the same. Hence: 
Theorem :— 
If any simple closed curve. x = p (t), y = > (©), can be divided into a 
finite number of arcs, of type y = f (x) f (x) being continuous, and of lines 
parallel to 0 y, and if intersections of x = & not meeting an end p int of 
an arc y = f (x) but meeting C be arranged according to the magnitude of 
their ordinates, the senses of description of the arcs at these points of inter- 
section are alternately positive and negative, and the sense at the inter- 
section of greatest ordinate is independent of &. 
This is the result that we have made use of in the second chapter. 
