776 THE THEORY OF TRANSFORMATIONS [ch. 



compensated for by the extent to which it has also got flattened 

 or thinned. In short, if we could permit ourselves to conceive 

 of a haddock being directly transformed into a plaice, a very 

 large part of the change would be simply accounted for by supposing 

 the former fish to be "rolled out," as a baker rolls a piece of dough. 

 This is, as it were, an extreme case of the balancement des organes, 

 or "compensation of parts." 



Simple Cartesian co-ordinates will not suffice very well to 

 compare the haddock with the plaice, for the deformation under- 

 gone by the former in comparison with the latter is more on the 

 lines of that by which we have compared our Antigonia with our 

 Polyprion; that is to say, the expansion is greater towards the 

 middle of the fish's length, and dwindles away towards either 

 end. But again simplifying our illustration to the utmost, and 

 being content with a rough comparison, we may assert that, 

 when haddock and plaice are brought to the same standard of 

 length, we can inscribe them both (approximately) in rectangular 

 co-ordinate networks, such that Y in the plaice is about twice 

 as great as y in the haddock. But if the volumes of the two 

 fishes be equal, this is as much as to say that xyz in the one case 

 (or rather the summation of all these values) is equal to XYZ 

 in the other; and therefore (since X = x, and Y = 2y), it follows 

 that Z = z/2. When w^e have drawn our vertical transverse 

 section of the haddock (op projected that fish in the yz plane), we 

 have reason accordingly to anticipate that we can draw a similar 

 projection (or section) of the plaice by simply doubling the ^'s 

 and halving the e's : and, very approximately, this turns out to 

 be the case. The plaice is (in round numbers) just about twice 

 as broad and also just about half as thick as the haddock; and 

 therefore the ratio of breadth to thickness (or y to z) is just about 

 four times as great in the one case as in the other. 



It is true that this simple, or simplified, illustration carries us 

 but a very little way, and only half prepares us for much greater 

 complications. For instance, we have no right or reason to pre- 

 sume that the equality of weights, or volumes, is a common, 

 much less a general rule. And again, in all cases of more complex 

 deformation, such as that by which we have compared Diodon 

 with the sunfish, we must be prepared for very much more 



