54 ROYAL SOCIETY OF CANADA 
IV: Regional Conditions : 
CON ONE La ae : 
Mb di TON TO ATEN 
V: Jsoperimetric Condition : 
gt D y (yx — ay"). dr = a 
TO 
S3. Reduction to an Non-isoperimetric Problem. 
(a). We are now going to reduce our problem to the non-isoperi- 
metric type oy means of the transformation: 

6 PME à 
DD = EE V(t) = fy Qa) ny) de ee (11) 
The quantities, U and V, have a simple geometric meaning, viz., 
D Cosy: 
(see fig. 6) and Vis the volume obtained by 
M revolving the sector, OMP, about the x axis. 
P The transformation, (12), co-ordinates with 
every curve, £, in the (x, y) plane a curve, 
£’, in the (U, V) plane, whose properties we 
(Fig.6) have to study.’ . 
(b). The function, r (t) 
By (LV: b), and (III: b), we have 
PCO) o>) OP ON FE cree secret sa eats i ote tee (12) 
Since g and # are of class D’, and r > 0 for 7, < T7 < 7,, we have that 
r (7) 1s of class D'or 7, < 7 Xa, and we have 
ga a HER inane ck (13) 
we 
bie ir 
L 
Ja — 
an equation that remains true at the points, 6, if we understand by these 
the progressive (regressive) derivatives. Now, from (III: a), as 7 = 
To, © approaches a definite limit, finite or infinite. Since y > 0 for 
7 
Ti To LT < Ty We may write, 

1 The discontinuities of 9’ and ¥’ we call the points, 6, 
* The results of this discussion will be found tabulated on p. 59. 
