XVII] THE COMPARISON OF RELATED FORMS 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 "reniform" 

 shape to precisely the same phenomenon, namely, to the existence 

 of a node or "hilus," about which the forces of growth are radially 

 and symmetrically arranged. 



Most of the transformations 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 transformations, on these general 

 lines, provide for the cases of a coaxial system where the 

 Cartesian co-ordinates are replaced by coaxial circles, or a con- 

 focal 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 Y — AX = c corresponding to the 

 logarithmic spiral 6 — A log r = c (Fig. 361). This beautiful and 

 simple transformation lets us at once 

 convert, for instance, the straight 

 conical shell of the Pteropod or the 

 Orthoceras into the logarithmic spiral 

 of the Nautiloid ; it involves a math- 

 ematical symbolism which is but a 

 shght extension of that which we 

 have employed in our elementary 

 treatment of the logarithmic spiral. 



These various systems of co- 

 ordinates, which we have now briefly ^. „,, 

 . . Fig. 361. 

 considered, are sometimes called " iso- 

 thermal co-ordinates," from the fact that, when employed in 

 this .particular branch of physics, they perfectly represent the 

 phenomena of the conduction of heat, the contour lines of equal 

 temperature appearing, under appropriate conditions, as the 

 orthogonal lines of the co-ordinate system. And it follows that 



