82 ROYAL SOCIETY OF CANADA 
to the representation of the curve, C, as a function of ¢ we wish to show 
that the senses of two arcs are the same or different according as the species 
are the same or different. To see this it will only be necessary to show 
that as we vo from one arc, C,, of type, y = f, (x), to the succeeding one, 
Cs, y = f, (x) neither sense nor species changes, or both. If x is an 
increasing function of t on C, and Q,, C, and (’, cannot have a common 
end-point ; for if they did, C, and C, together would form a single are of 
. type, y = f (x). Hence C, and C, must be joined by a straight line 
parallel to 0 y; from A (x,, y,) say to B (x,. y,') (fig. 18). Select any 
points À (&, 7), and S (&' 7’) on C and C, respectively such that 
_ 
| fi (a) TE fi (7) | << gis mth for ay SS: F4 
fa) — a) eee EE 

Let m be any positive quantity less than the least distance of RABS 
from the curve C with C,, AB, C, deleted. Let € be some positive 
quantity, € << m,«,— & & — x. Then the line æ = x, — € between 
{a ea fi CG pe e) | and Yahi De Yas does not meet C except at 
a 2 
1% Se Wer CNE RE a}. Similarly the line, + = a, + € between 
B S S D 

{ 
| 
{ 
| 
Ke 
(Fig.19) 

A 
(Fig. 18) 
te + €, fi (an + e) l'and ne does not meet C except at 
1% +f, @+ 8) L. Since 
Bs a — E <a, And a LE + e < E, 
the straight line joining M, (a -—e, Po and M, (a, + &, wry") 
does not meet C, or C,, and since €< m, it does not meet any arc of C other 
than AB. The middle point of M, M, being on AB, must have points of 
classes 2 and 3 in every vicinity, however small. It follows easily that 

Dini er reno: 
