be, as it were, inverted, the broad, more or less heart-shaped, outline appearing at the upper end, while below the leaf tapers gradually downwards to an ill-defined base. In the latter case, as in Dionaea, we obtain a leaf equally expanded, and similarly ovate or cordate, at both ends. We may notice, lastly, that the shape of a solid fruit, such as an apple or a cherry, is a solid of revolution, developed from similar curves and to be explained on the same principle. In the cherry we have a ãpoint of arrestã at the base of the berry, where it joins its peduncle, and about this point the fruit (in imaginary section) swells out into a cordate outline; while in the {735} apple we have two such well-marked points of arrest, above and below, and about both of them the same conformation tends to arise. The bean and the human kidney owe their ãreniformã shape to precisely the same phenomenon, namely, to the existence of a node or ãhilus,ã about which the forces of growth are radially and symmetrically arranged.
Most of the transôÙforôÙmaôÙtions which we have hitherto considered (other than that of the simple shear) are particular cases of a general transformation, obtainable by the method of conjugate functions and equivalent to the projection of the original figure on a new plane. Appropriate transôÙforôÙmaôÙtions, on these general lines, provide for the cases of a coaxial system where the Cartesian co-ordinates are replaced by coaxial circles, or a confocal system in which they are replaced by confocal ellipses and hyperbolas.
Yet another curious and important transformation, belonging to the same class, is that by which a system of straight lines becomes transformed into a conformal system of logarithmic spirals: the straight line Yÿ£¢ãÿ£¢Aÿ£¢X =ã₤c corôÙreôÙsponôÙding to the logarithmic spiral ö¡ã₤ãã₤Aã₤logã₤r =ã₤c (Fig. 361). This beautiful and
simple transformation lets us at once convert, for instance, the straight conical shell of the