216 
Tt then follows from merely mechanical reasons that the 
tangential tension is the same at all points, and that the form 
of the portion of the margin at the instant under consideration 
is determined by one of the following intrinsic equations: 
2s . 
s Z 7 cB sin (8+ ¢) 
tan (7-008 2) =cos a. tan $, or é = gin (6 ~ 4) 
according as the vigour of growth is more or less than sufficient 
to overpower the direct resistance of the current. 
In these equations s represents the length of the are of the 
margin measured from that point of it where its tangent is in 
the direction of the current to the point where the tangent is 
inclined to that direction at the angle @; and J and a or B are 
quantities dependent only upon the proportional values of the 
tangential tension, the power of growth and the direct resist- 
ance of the current. The first equation when traced furnishes 
a series of separate ovals (but not ellipses) the longest diameters 
of which all lie on one straight line perpendicular to the direc- 
tion of the current; the secoud equation furnishes a pair of 
catenary-like curves with their convexities opposed to each 
other, which become actual catenaries when the power of growth 
would just balance the direct resistance of the current. Parts 
only of these curves are applicable to the case of the portions 
of the leaf-margins according to the original hypothesis, and in 
no case are those parts of the curves applicable which corre- 
spond to pomts where ¢ lies between 180° and 360°. 
After the leaf-margin ceases to be flexible, as for imstance 
after the completion of its growth, the investigation can be 
extended to calculate the tangential tensions, the normal strains 
and the wrenching couples to which it is then submitted at the 
different points of the margin; and tolerably simple expressions 
are found for them. 
The above equations are only suitable for those leaves in 
which the structure is pretty uniform in all directions, as in 
