FORMS AND DISTRIBUTION OF BATRACHIUM. 107
1871, before the Cambridge Philosophical Society : — Consider the curved margin of an undivided portion of a leaf which floats in a stream exposed to the resistance of the current ; suppose that the power of growth is exerted equally at all points of the margin, and tends to push the margin normally outwards, so as to oppose rather than co-operate with the cur- rent, and is just balanced at the instant considered by the other mechanical forces which act on the margin ; and further suppose that the margin remains as a flexible curve with tangential tension, but not sul)mitted to either normal strains or wcenching couples. It 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 con- sideration is determined by one of the following intrinsic equations : —
��(^ycosajn
��tan I -r cos a f zz cos o tan (p, or e
��2s.cotj3 I sin (3 -f .
��sin (3 ~ (p)
��according as the vigour of growth is more or less than sulRcient to over- power the direct resistance of the current.
In these equations * represents the length of the arc of the margin measured from that point of it where its tangent is in tlie direction of the current to the point where the tangent makes the angle </> with that direction ; and I and a or /3 are quantities dependent only upon the pro- portional values of the tangential tension, the power of growth and the direct resistance of the current.
It readily follows from these equations that the curvature of the mar- gins, at those points where the tangent makes a small angle with the direction of the current, is greater than at those points where the tangent makes a larger angle. After the leaf-margin ceases to be flexible, as, for instance, after the completion of its growth, the investigation can be extended to calculate the tangential tensions, the normal strains and the wrencliing couples, to which it is then submitted at different points of the margin, and tolerably simple expressions are found for them. 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 direction of the current ; the second 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 ap- plicable to the hypothesis ; and in no case are those parts applicable which correspond to points where tfy lies between 180° and 360°.
When the leaves are divided, as is frequently the case, each lobe must be treated to a separate calculation ; and when the margins are exposed to violent strains or abnormal mechanical conditions, growth is pro- bably checked and the leaves tend to retain their form by the support of their interior, but new lobes may be produced at those points where the tendency to break, as determined by the method above indicated, is a maximum.
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