Pteropod or the Orthoceras into the logarithmic spiral of the Nautiloid; it involves a mathôÙeôÙmatôÙiôÙcal symbolism which is but a slight extension of that which we have employed in our elementary treatment of the logarithmic spiral.
These various sysôÙtems of co-ordinates, which we have now briefôÙly conôÙsiôÙdered, are someôÙtimes called ãisoôÙtherôÙmal co-ordinates,ã from the fact that, when emôÙployed 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 {736} the ãlaw of growthã 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 mathôÙeôÙmatôÙiôÙcal transformation.
When the system becomes no longer orthogonal, as in many of the following illustrationsãfor instance, that of Orthagoriscus (Fig. 382),ãthen the transformation is no longer within the reach of comparatively simple mathôÙeôÙmatôÙiôÙcal analysis. Such departure from the typical symmetry of a ãstream-lineã 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 mathôÙeôÙmatôÙiôÙcal 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 ãlaw of growth,ã or play of forces, by which the transformation has been effected.
Before we pass from this brief discussion of transôÙforôÙmaôÙtions in general, let us glance at one or two cases in which the forces applied are more or less intelligible, but the resulting transôÙforôÙmaôÙtions are, from the mathôÙeôÙmatôÙiôÙcal point of view, exceedingly complicated.
The ãmarbled papersã of the bookbinder are a beautiful illustration of visible ãstream lines.ã On