[WILSON] CERTAIN TYPE OF ISOPERIMETRIC PROBLEM 88 
HU T Se, ‘ : 
Ve (ya — ay’), whence the volume may be taken as defined by : 
dt 
uf AT Ti ; , 
V = ps . y (yx — ay’) . dr. 
Summing in asimilar fashion the attraction integral takes the form : 
A= Py % (yx' — xy’) . dr, 
where r = 4/a* + y, and x is a constant factor uepending upon the 
constant of gravitation, Dropping constant factors, we have to find a 
‘T) ‘ 
maximum value for J Bay 7 GS) Irene proto ot ece (9) 
F 
To 
subject to the condition, 
Kea Ue y (yx' — ay’) ENGI On ee eee ey Dear (10) 
TO 
c) being a certain positive constant. 
(c). Stating these hypotheses arithmetically, and collecting them 
for purposes of reference, we propose to find a maximum for the 
integral J, = Vo . (yx' — xy') dr for the admissible curves, 
oi 
Lie = p(t) y = p (Tr) in (Ty 
where I: General Characteristics : 
(a) and + are of class D’ on (7, 7,) ; 
(b) p (%2) = P (7), and  (7,) =  (7,), cannot both be true if 
MTS 
Il: Slope Condition: ya' — xy' = 0, 
(a) at a finite number of pointe, 7 = y; 
(b) on a finite number of segments, «, < T7 < AI ASS 
IIL: Initial Conditions : 
(a) © (ro) = 0, % (7%) = 0, ? (ri exists ; 
; (7) TO + 0 
I 

(b) YP (Ti) = 0, ¢ (Ti) = 0; 
