[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 67 
drawn not meeting an arc, 6 = const., of this curve, then the points of 
intersections of the ordinate with the curve are finite in number, and if these 
intersection be arranged according to increasing values of r, 0 is alternately 
a decreasing and increasing function of r. Further for all values of 6, Bis 
a decreasing function of * for the intersection for which r is the greatest. 
(d). To see the truth of the last statement, we observe that P (7,) can- 
not be the end-point of a segmention which # =0, For if it were, since 
6 (7) = 0, we should have 6=0 on this segment, contrary to (2). 
ome very is an end-point of an are of type, r = f, (4). Let 7 = 7, be the 
other end-point. Since 6 (r,) = 0, we have from (2) 
i SOS CR) eu) (ay) 
Hence 4 (7) is a decreasing function of t onr =f, (0). In the same 
Way, if 0 (To) = 0, (fig. 10), we may show that Q is an end- -point of an 
are on which r = f, (4); let +,’ be the other end-point. If 6 (HERO! 
the distances, 9 (+), of the are, P Q, for t, <1 <1, from 0 r have a 
minimum.’ Since P Q meets 0 r only at _ (7 ss" we in the first case | 
and ouly at P and Q in the second, (* > 7,7 <7 > )» these minima are 
positive, = 2m say. Then if 0 Le <m, ot ie, PIE MOMIE TRACE) 
(drawn from 2 in figs. 9, 10), meets our curve at { UE) ke and at 
this point only. For since 6 = fr (e) > 0, (see eqn. 2), it does not meet 
Oror0 6. Since & <°m, it does not meet any arc of PQ other than 
r =f, (9), and possibly r = f, (4) if 6 (7,) = 0. It can meet r = f (6) 
at butone point, viz., R. If it meet r = f, (#), we should have fo (A) — 
f, (2) > 0 for 0 =E Nowf, (6) — f, (6) <0 for 0 = 0. Hence since 
fo (8) and f, (9) are continuous, (see (c)) f, (4) — f, (0) =0 for some 6, 
0<@<&, This contradicts what we have proved in (b), viz., that the 
arc, P Q,is simple. Hence the given halfline, 6 = Enea CS), re not 
meet our curve. Hence for 0 <¢<m, the greatest value of r for the 
intersections of 6 = eis onr = f, (4). °We have shown that on this are, 
@ is a decreasing function of r. We shall prove that the sense of 4 
for the greatest r is independent of #’. Hence 6 is always a decreasing 
function of for the intersection for which r is that the greatest. 
(e) In § 3 (a), we have transformed the curves, £, into the curves, 
£”, and have the relations, 
UT pe COSMO UN ea BS" OUR DA OU Na REC (3) 
(see § 3 (a), and § 3 (e), eqn. 20), where P (+) is the slope of the new 
curve. The arcs, @ = const., (i.¢., y x’ — x y’=0), become stationary 

— 
' Dini Funktionen, s. 68. 
2 Dini, 1.c., 8 70: 
* These results of Analysis Situs are proved in the third chapter. 
