868 D'ARCY WENTWORTH THOMPSON ON 



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 the "law of growth" which our biological analysis, by means of orthogonal 

 co-ordinate systems, presupposes, or at least foreshadows, is one according to which 

 the organism grows or develops along stream lines, which may be defined by a 

 suitable mathematical transformation. 



When the system becomes no longer orthogonal, as in many of the following 

 illustrations — for instance, that of Orihagoriscus (fig. 38), — then the transformation is 

 no longer within the reach of comparatively simple mathematical analysis. Such 

 departure from the typical symmetry of a " stream-line " system is, in the first 

 instance, sufficiently accounted for by the simple fact that the developing organism 



Fig. 10. 



is very far from being homogeneous and isotropic, or, in other words, does not behave 

 like a perfect fluid. But though under such circumstances our co-ordinate systems 

 may be no longer capable of strict mathematical analysis, they will still indicate 

 grajihically the relation of the new co-ordinate system to the old, and conversely 

 will furnish us with some guidance as to the " law of growth," or play of forces, by 

 which the transformation has been effected. 



Before we pass from this brief discussion of Transformations in general, let us 

 glance at one or two cases in which the forces applied are more or less intelligible, 

 but the resulting transformations are, from the mathematical point of view, ex- 

 ceedingly complicated. 



The " marbled papers " of the bookbinder are a beautiful illustration of visible 

 " stream lines." On a dishful of a sort of semi-liquid gum the workman dusts a 

 few simple lines or patches of colouring matter; and then, by passing a comb 

 through the liquid, he draws the colour-bands into the streaks, waves, and spirals 



