[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 81 
(4,,7,). Then a = &, meets only a firite number of arcs of type, y = f(x). 
If there be any such that the ordinate > 7,, let 7,’ be the least of these; 
then 7,’ > 7, The point-set, (&, y), such that 7,’ > y > 7 for any &, 
within (x, #,). we call the points immediately above C, Similarly we 
detine the points immediately below C, by 
ny <m for any &, within (x, 2), 
where (7,”, &,) is the intersection of x = £,, of greatest ordinate less than 
yn, If there are no intersections, (£, y) such that y > 7, (or y <7,), we 
take 7,’ = 0 (or 7,” = — o). 
(b). The points immediately above C, are all of the same class. For let 
P (&,, y,) and Q (&:. yz) be any two points immediately above C,. Then, 
from the definition, if (&,, 77,) and (&,, 7,) be the points at which x = &, 

and x = &, meet C,, these points are interior to C, Let m be the least 
distance between C, from P to Q (inclusive) and the remainder of the 
curve, C, excluded. Then m> 0.' Select any positive 06, d <m, y,— 7, 
Y2 — M and construct the curve, y = f, (x) + 6, between a where x = &,, 
and b where x = &. Since 6 < m, this cannot meet the curve C. Then 
Pab Q forms a continuous curve not meeting Cand joining P and Q. 
These, therefore, belong to the same class. Similarly points immediately 
below C are all of the same class. Since there must be points of both 
‘classes in every vicinity of points on C, (see § 1), the points immediately 
above C, and immediately below C are of different classes, 2 and 3. 
(c). We say that an arc, C,, of type y =. (x) is of species 2 or 8 
according as the points immediately above it are of class 2 or 3. Returning 

1 Dini, Grundlagen, etc., s. 68. 
Sec. IIT., 1907. 6. 
