pair of equal catenary-like curves is obtained with their con- 
Vexities opposed to each other, which in the last case, when 
1 
p=5 ov’, become actual catenaries, 
Those parts of the curves obtained above which correspond 
to points from ¢=0 to @=7 are alone applicable to the 
original hypothesis that the resistance due to the current 
opposes the power of growth; parts corresponding to points 
from @=7 to @=2r are either removed from the influence 
of the current (when circular ares are obtained for such portions 
of the margin), or if subject to the influence of the current (by 
a slight obliquity in the plane of the leaf, the anterior margin 
being depressed), would have it so as to assist rather than 
oppose the power of growth, and the equation for such portion 
of the margin would be 
ds T 
db wr F 
$ Pts sv’ sin" 
3 
ov 
assume tan*y = —., 
2p 
tar 
then =f ER 
1 ae f. y.sin'd J 14sec" y. tan’ 
=cos y. tan™. (see y. tan ¢), 
p Sil ng 1 
2= — = ——.. tan™, (tan y. sin 
T 1+t a y.sin'd — tany Coy 
p d.cosd cos" y 1—sin y. cos | 
-.YyoSe | FS SS —_—————__—— 
rd sec'y—tan*y.cos’'@ 2siny’ ‘sad 1+siny. cos f° 
siny » siny ru 5) 
—— ey vy t2 
therefore eces*y't "4 @ costy’r” = COS, tany.2 2) : 
cos y T 
