[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 65 
(f) Since £’’ is continuous, we have 
v=ec' w+ ec” 
throughout. Since it must pass through (u,, 0), and (1, @), its equation is 
v (us — ONE (up: ~ à 
- > 0,* and is finite, u, = 1- Computing 7”, we get 
u 
oS eat us )s. 
This is evidently a maximum when Us = 0; in this case 
Since 
7 Je 
Vo 
The maximising eurve, if any exist, is therefore given by : 
S 7. Slope Properties of Curves, £. 
(a) In order to prove that the curve, ©, obtained at the conclusion 
of the preceding paragraph, actually furnishes a maximum, we shall 
need certain results with reference to the slope properties of the ares, £v’’, 
These are connected with certain “outside, inside and outside,” properties 
of the original solid, and are most easily_deducedeby/returning to the 
original set of curves, £, (see §2(c)). We represent these in an (r, 6) 
plane by means of the transformation, 

| 
PF + Eu Zoo cg eT, docibe aera bent: (1) 
aw 
0 rand 0 4 being a pair of rectangnlar"axes. Since # > 0, (see IV: b)? 
for any 7, KT < T» Ÿ is uniquely determinate for any such 7. As 
ph a approaches a determinate limit, (see ITT:a)*, positive if a. 
p 17 +0 
be finite, and 0 if infinite. Ag r — 7, Since @ (7,) > 0, and # Co yaaa 
Sp tpproaches infinity (III: b), and therefore 4 == 0, For the point, 
1 See §5:b(A:3). 
* These references are to the tabulated results, $4 (c.) 



Sec. III., 1907. 5. 
