1088 THE THEORY OF TRANSFORMATIONS [ch. 



It is obvious enough that the same method of description and 

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

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

 error, our various cross-sections arid the coordinate 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 prin- 

 ciples of such correlation, to forecast approximately what is likely 

 to take place in the third dimension when we are acquainted with 

 two, 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 coordinates as 

 some which we have studied, owing to the presence of essentially 

 unimportant, but yet conspicuous differences in the position of the 

 eyes, or in the number of the fins — that is to say in the manner in 

 which the continuous dorsal fin of the plaice appears in the haddock 

 to be cut or scolloped into a number of separate fins. But speaking 

 broadly, and apart from such minor differences as these, it is manifest 

 that the chief factor in the case (so far as we at present see) is simply 

 the broadening out of the plaice's body, as compared with the 

 haddock's, in the dorso-ventral direction, that is to say, along the 

 y axis; in other words, the ratio x/y is much less (and indeed little 

 more than half as great) in the haddock than in the plaice. But 

 we also recognise at once that while the plaice (as compared with 

 the haddock) is expanded in one direction, it is also flattened, or 

 thinned out, in the other: y increases, but z diminishes, relatively 

 to X. And furthermore, we soon see that this is a common or even 

 a general phenomenon. The high, expanded body in our Antigonia 

 or in our sun-fish or in a John Dory is at the same time flattened 

 or compressed from side to side, in comparison with the related 

 fishes which we have chosen as standards of reference or comparison ; 

 and conversely, such a fish as the skate, while it is expanded from 

 side to side in comparison with a shark or dogfish, is at the same 

 time flattened or depressed in its vertical section. We hasten to 

 enquire whether there be any simple relation of magnitude dis- 



