774 THE THEORY OF TRANSFORMATIONS [ch. 



In this brief account of co-ordinate transformations and of 

 their morphological utility I have dealt with plane co-ordinates 

 only, and have made no mention of the less elementary subject 

 of co-ordinates in three-dimensional space. In theory there is 

 no difficulty whatsoever in such an extension of our method; it 

 is just as easy to refer the form of our fish or of our skull to the 

 rectangular co-ordinates x, y, z, or to the polar co-ordinates 

 I, 7], I,, as it is to refer their plane projections to the two axes to 

 which our investigation has been confined. And that it would 

 be advantageous to do so goes without saying ; for it is the shape 

 of the solid object, not that of the mere drawing of the object, 

 that we want to understand ; and already we have found some 

 of our easy problems in solid geometry leading us (as in the case 

 of the form of the bivalve and even of the univalve shell) quickly 

 in the direction of co-ordinate analysis and the theory of conformal 

 transformations. But this extended theme I have not attempted 

 to pursue, and it must be left to other times, and to other hands. 

 Nevertheless, let us glance for a moment at the sort of simple 

 cases, the simplest possible cases, with which such an investigation 

 might begin ; and we have found our plane co-ordinate systems 

 so easily and effectively applicable to certain fishes that we may 

 seek among them for our first and tentative introduction to the 

 three-dimensional field. 



It is obvious enough that the same method of description and 

 analysis which we have applied to one plane, we may apply to 

 another : drawing by observation, and by a process of trial and 

 error, our various cross-sections and the co-ordinate systems 

 which seem best to correspond. But the new and important 

 problem which now emerges is to correlate the deformation or 

 transformation which we discover in one plane with that which 

 we have observed in another: and at length, perhaps, after 

 grasping the general principles of such correlation, to forecast 

 approximately what is likely to take place in the other two planes 

 of reference when we are acquainted with one, that is to say, to 

 determine the values along one axis in terms of the other two. 



Let us imagine a common "round" fish, and a common "flat" 

 fish, such as a haddock and a plaice. These two fishes are not as 

 nicely adapted for comparison by means of plane co-ordinates as 



