88 ROYAL SOCIETY OF CANADA 
Hence the problems of finding the isoperimetric maximum of I for the 
totality of curves, £, and of finding the ordinary maximum of I’ for the 
totality of curves, £', are equivalent. 
$4. Removal of Stationary Points.’ 
(a). We have remarked in $3 (a) that the segments, ou which 
yx' — wy’ = 0 are straight lines whose direction passes through the 
origin. From the definitions of U (7) and V (7) given by the equations 
(11), it follows that these segments in the parameter, 7, correspond to 
stationary points on the curve, £’, given by (22). In order to eliminate 
these we introduce a new parameter, ¢, by means of the following 
substitutions : 
7 = ?¢, for 7 on the first interval not a x A—interval, 
Tr = t + d, for7 on the second interval not a « A—intervil, 
7™=t+d,+ 4, for 7 on the third interval not a « À —interval, 
where d, = A, — «; , i being determined as follows: 
I 
1) when; js not an end-point of a ,A—interval,i = 1,2, 5 . . .; 
0 
D LC 
5 Ae 
ii) when - is not an end-point of a « A—interval, à 
Geometrically, we delete the segments of the parameter, 7, corresponding 
to the stationary points of £’, and in order to remove the resulting gaps 
we make a simple translation of the parameter. We suppose that the 
new parameter, f, has the range, ¢,, f;, and denote by 4,, k,, k,, the values 
of ¢ at the points of junction, viz: 
GQ) Ay = 3 hy = KH, — di; hy = ey à — de; 
(it) ky =k,; kp = Ky dd; kg = eg — d — Fe; 
With these relations we detine: 
u(iy= UN); 0 @) = PAG) D @) SF Gener ee (23) 
It follows that the curves, 
= TNT 0 = Aa) Mort ONNA Gr D, 
and UN UN(E) BY =) 0 (CE), Moro AE 
are identical as locus curves, and p (t) is the slope of the latter. In case 
(ii), and this case only, p (@¢,) = P (r) = 0by ([V':b). Further) 
p (kj — 0) = P («,),or P (ki 4 1), and p (k; + 0) = P (Aj) or P (Aj 4 1) 
in cases (i) and (ii) respectively. Since V’ (7) = 0 on (x, À,) in the last 
case from (III’: a), V (A,) = v @,) = 0. 
It is to be noted further from the values of U’ and V' given by 
equations (15) and (16) that the segments, («; À; ), contribute nothing 
to the integral, 
T1 
% 24 
se de D NN MUNIE 

1 The results of this discussion are given on p. 14. 

