[WILSON] CERTAIN TYPE OF ISOPHRIMETRIC PROBLEM 71 
vicinity (7 7”), 7” > + If is not a point of regression or an end-point, 
u (t) is an increasing (decreasing) function of t at 7; 7.¢., it is increasing 
(decreasing) in some vicinity (r” 7”), 7 << r < 7 Since v, < v < v, 
u = v does not meet a point of regression on £* or an end-point. Hence 
u (t) is an increasing (decreasing) function at each intersection of u = v 
and £*, The number of these is finite. Jf 7 be the least among their 
paramrters, then u (t) is an increasing functionoftat7, Foru, <v, <v; 
it follows that v = u (7) > wu, = u (tf). If u (t) were a decreasing 
function at 7, there would exist a ¢’, t’ <7, such that the difference, 
u (7) — u(t), positive at t,, and negative at ¢t’, must vanish for some 7’, 
to LT LE. This contradicts the hypothesis that + is the least among 
the values of ¢ for which w (ft) = v. Inthe same way we have that at 
the intersection with the greatest parameter, since v < v, <1, u (€) is 
an increasing function of t; a d that, arranged in order of their para- 
meters, at the intersections « (¢) is alternately increasing and decreas- 
ing. Hence the total number of intersections of £* with u = v must be odd. 
(c). Slopes on £*. 
We have seen, ($7 f, g), that if we number the arcs between v, and 
Ua, 1,2,3 . . . . . according to magnitudes of the slopes at the inter | 
sections of £ and u = v, the numbering is independent of v. The same 
is true of £*, For since £ and £* coincide except for the straight 
line u_, Uo, this will certainly be true for any interval (v, v,) that does 
not contain points of (w_, u,) on its interior. If it does contain such, the 
order of the slopes on the arcs of £* other than (uw, uo) will be fixed, 
Since p*= 0 onu, u, and p> 0 within (v, v,), (IV”: b, p. 15), the slope 
will be less on uw_, u, than on any other arc, and the order is still fixed. 
We have then between v, and ».. 
Die Dale) Das leet visi Nr PAR RTE ES CL) 
At the points, v, and v,, some of the signs of inequality may be replaced 
by signs of equality if we understand by p, (v,) the value of p, (v, + 0) 
and by p, (v2), the value, p, (v — 0). 
(d). Construction of C. 
We now construct a curve, ©, whose slope, p, is given by 
DUR D Agta NON ps NN ie esse (2) 
on (v1 v,), which shall be continuous, and pass through (w, , 0) ; v.e., 
C'(d, u), where: i [3 . du. 
u-1 
