1048 THE THEORY OF TRANSFORMATIONS [ch. 



contour lines of equal temperature appearing, under appropriate 

 conditions, as the orthogonal lines of the coordinate system. And 

 it follows that the "law of growth" which our biological analysis 

 by means of orthogonal coordinate 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 Orthagoriscus 

 (Fig. 526) — 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 coordinate systems may 

 be no longer capable of strict mathematical analy'sis, they will still 

 indicate graphically the relation of the new coordinate 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 compHcated. 



The "marbled papers" of the bookbinder are a beautiful illustra- 

 tion 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 Hquid,.he draws 

 the colour-bands into the streaks, waves, and spirals which con- 

 stitute 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 



