[ WILSON] >JERTAIN TYPE OF ISOPERIMETRIC PROBLEM 59 
Tt follows that 
| PANDA EE 
(GNU AR ENS 
to 
It remains to enunciate the conditions given in (4) of the preceding para- 
graph in terms of w (t) and v (t). 
(b). In the first place we have from (/’) that v (1) is of class D’ on 
(to t,), and wu (t) is of class C on this interval, and except perhaps for 
t = t,, of class D’ on the same interval. From (II’:a), w’ (t) and v’ (1) 
can vanish at only a finite number of points, # = g. The fanction, p (ft), 
will not, in general, be continuou:; from (II’: b), however, it must be of 
class D' on the intervals, (t, &,), (A, hz), (&, t,), except perhaps for 
t=t, In (ii) of the preceding paragraph, as 7 covers the interval, 
x, A,, P (7) takes all values from 0 to P (A,) = p (¢,), inclusive. Hence 
CII’ : ¢,) is the equivalent of (II’:c¢) in this case. [n the remaining cases 
as t describes the interval, («;4;), P (7) takes all values between 
p (k; - 1 —9) and p (Aj_1 + 0) or p (k, — 0) and p (k; + 0) accord- 
ing to whether it belongs to (i) or (ii); a remark which makes 
(II’’: ¢,, ¢,) the equivalent of (II: c) in these cases. If we write u (¢,) 
=u, it follows from ([V’) that in (ii) of the preceding paragraph, Le. 
when p (t,) = 0,u,<1. In this case also it follows that from (I’: b) 
that u (¢) is of class D’ up to and including ¢ = ¢,.. The remaining con- 
1? 
ditions are translated easily. 
(c). Collecting these results, the family of admissible curves, £’ in 
the (u, v)—plane transforms into a set of curves included in the totality, 
£”, with the following properties : 
I’: General Characteristics : 
(a) v (¢) is of class D’ on (¢, t,); 
(b) u (2) is of class C on (@, 7,), of class D’ fort, Lt <t, and 
if p(s), se for tt; 
0? 
Il”: Slope Conditions : 
(a) v’ and wu’ vanish at only a finite number of points, { = y; 
Che) stor elass Wonk iC Nha EN ONCE 
p = — where this quotient is defined ; 
u 
(¢,) p (2) = p(1,) and u (7,) = u (f,) cannot both be true af 
ty Lt; 
(c,) if u (t,) = u (K; ), then p (t,) cannot lie between p (k — 0) 
and p (k +0); 
(es) ifu (f,) = u,, then p (t,) > p (t,) unless £, = À ; 
oO 
