XVII] THE COMPARISON OF RELATED FORMS 1047 



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 coordinates 

 are replaced by coaxial circles, or a confocal system in which they 

 are replaced by confocal ellipses and hyperbolas. 



Fig. 504. Outline of a compound leaf, 

 like a horse-chestnut, based on a 

 composite sine-curve, of the form 

 r = sin 012 . sin nd. 



Fig. 505. 



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 

 ^ — ^ log r = c (Fig. 505). This beautiful and simple transforma- 

 tion 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 mathematical symbolism which is but a 

 slight extension of that which we have employed in our elementary 

 treatment of the logarithmic spiral. 



These various systems of coordinates, which we have now briefly 

 considered, are sometimes called "isothermal coordinates," from the 

 fact that, when employed in this particular branch of physics, they 

 perfectly represent the phenomena of the conduction of heat, the 



