60 ROYAL SOCIETY OF CANADA 
IIL”: Lnitial Conditions : 
(a) v @,) = 0, (4) = ©; 
(b) u@,) =u, u (4) = 1, Ui l'unless p (t,) = 0; 
ae: 6 %# 
(CO) eer ea aNd ONCE / exist and are finite ; 
V1 — wi F0 V1 — zl te 10 
IV": Regional Conditions : 
(COMORES MT a an 
(D) pit) b> 0 forut acpi if 
The integral to be maximised, 
MBE 
r= f ROG Raya ans a 
oO 
Conversely, if any curve, £’, of the totality just defined, be given, and 
we transform the parameter, t, in a manner entirely the inverse of that 
used in (i), it is not difficult to see that we obtain a curve of the totality 
er 
À 
QUE 74 
» Such that jo Vv’. U'.U . dr along the curve, £’, is equal to 
To 
BUS 
a v'.u'.u. dtalong the given curve. It follows that the problems 
lo 
of finding a maximum for I’ among the curves, £’, and a maximum for J” 
among the curves, £’’, are equivalent. 
$5. Decomposition of £" into ares of type, v = f; (u). 
(a). We denote the points, g, at which v’ = 0, or wu’ = 0, the 
points, k; at which p (t) is discontinuous, and the points, 6, at which 
p’, v'and wu’ are discontinuous, collectively by [a]; a (=t)) <a, < a, ..: 
<a@,41(=t,). The corresponding values of u are a, (=u,), 4, & ...@, 1 
(see IIL’: b}. We decompose any curve, £”, into ares, £;”’, and the in- 
terval, (?, t,), intosub-intervals at these points, 7 = 0, 1,2, ...7. Then 
on £," u’ (t) has a fixed sign, and wu’ (t) = 0 except perhaps at the end 
points. We may therefore solve for t as a function of u of class C on and 
C’ within £,’." Substituting in v (ft) and w (t), we obtain v and pas 
functions of u of class C on and C’ within £.’. 
; Now within £,” since 
u’ + 0 and is continuous, 

Dini, I CS: 
