[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 83 
M, and M, must be of different classes. Hence by the results at the end 
of (b) the points immediately above C, and the points immediately above 
C, are of the same class. Hence by the definition neither the class nor 
the species changes, as we go from C, toC,. Similarly if x is a decreasing 
function of t on C, and'C,. 
(d) If the senses on C, and C, are different, let ws suppose in the first 
place that x is increasing on ©; and decreasing on ©, As in (c) select 
an À and S, interior to ©, and C,, and a positive m less than the least of 
the distance from RABS, (see fig. 19) : À and B coincideif C, and C, are 
contiguous arcs. Then let ¢ be any positive quantity 
EP SO EEE BP Oh 
1 
Then the line, a = x — €, meets Ci at P À a — €, fi (a, — €) l , and 0, 
at Q {x — €, fs (& — €) and since & < m, it does not meet the curve 
otherwise. Points (a, — ¢, y), within PQ give f, (x, — ¢) — y, f, (a — €) 
— y, different signs, and hence are above one and below the other of 
C and ©; These are of the same class. Hence the points immediately 
above ©! and the points immediately above C, are of different classes. 
Similarly if x is a decreasing function of ¢ on C and increasing on C.. 
Hence when the sense of description with reference to 0 x changes, the 
species changes. We have thus the result of (b). 
(e) Draw any line, x = &, meeting C but none of the straight lines 
parallel to 0 y. Since the curve is simple, the intersections of « = & with 
the arcs of type y = jf (x) cannot coincide. Let them be arranged 
according to the increasing magnitude of their ordinates. Let R and S 
on (© and ©, be two snecessive intersections for this arrangement. 
Since À and S are successive intersections, RS can meet no arc of C onits 
interior. Hence points within RS are of the same class by the results of 
§ 1. As in (c), we may show that C, and C, are of different species. 
It follows from (c) that the senses of description of C, and C, are 
different. Further, since # (¢)' is continuous, we can obtain a b such 
that b > | > (t) | on C If theu 7 be the greatest of the ordinates among 
the intersections of x = & with C, the ordinate, x = &, does not meet C 
for y DIE OC 06 > | > (€), | this half-ordinate meets y = b. Similarly 
for any other value of x, x = &’. Since y = b does not meet C, points 
above the ares C,, (,, on which (7, &) and (7, &’) lie, which are joined 
by y = 6 not meeting C are of the same class. Hence from the definition 

1See p 26. 
? Osgood Funktionen theorie, s. 13. 
