736 THE THEORY OF TRANSFORMATIONS [ch. 



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

 orthogonal co-ordinate systems presupposes, or at least fore- 

 shadows, 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 Orfhagoriscus 

 (Fig. 382), — 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 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 graphically 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, exceedingly 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 which constitute the marbled pattern, and which he then 

 transfers to sheets of paper laid down upon the gum. By some 

 such system of shears, by the effect of unequal traction or unequal 

 growth in various directions and superposed on an originally 

 simple pattern, we may account for the not dissimilar marbled 

 patterns which we recognise, for instance, on a large serpent's 

 skin. But it must be remarked, in the case of the marbled paper, 

 that though the method of application of the forces is simple, 

 yet in the aggregate the system of forces set up by the many 



