MORPHOLOGY AND MATHEMATICS. 



879 



Chaetodonts will be found to correspond to very slight modifications of similar 

 co-ordinates ; in other words, to small variations in the values of the constants of the 

 coaxial curves. 



In figs. 33-35 I have represented another series of Acanthopterygian fishes, not 

 very distantly related to the foregoing. If we start this series with the figure of 

 Polyprion, in fig. 33, we see that the outlines of Pseudopriacanthus (fig. 34) 

 and of Sebastes or Scorpama (fig. 35) are easily derived by substituting a system of 

 triangular, or radial, co-ordinates for the rectangular ones in which we had inscribed 

 Polyprion. The very curious fish Antigonia capros, an oceanic relative of our own 

 " Boar-fish," conforms closely to the peculiar deformation represented in fig. 36. 



Fig. 37 is a common typical Diodon or Porcupine-fish, and in fig. 38 I have 

 deformed the horizontal co-ordinates into a system (approximately) of coaxial 

 parabolas. The old outline, transferred in its integrity to the new network, appears 

 as a manifest representation of the allied, but very different-looking, Sunfish, 



Fig 31. — Scar us sp. 



Fig. 32. — Pomacanthus. 



Orthagoriscus mola. This is a particularly instructive instance of deformation or 

 transformation. It is true that, in a mathematical sense, it is not a perfectly satis- 

 factory, or perfectly regular, deformation, for the system is no longer orthogonal ; 

 but nevertheless it is symmetrical to the eye, and obviously approaches to an 

 orthogonal system under certain conditions of friction or lateral constraint. And as 

 such it accounts, by one single integral transformation, for all the apparently 

 separate and distinct external differences between the two fishes. It leaves the parts 

 near to the origin of the system, or intersection of the main axes, such as the pectoral 

 fin, practically unchanged, and it shows a greater and greater apparent modification 

 of form as we pass from that origin towards the periphery of the system. In a 

 word, it is sufficient to account for the new and striking contour in all its essential 

 details, of rounded body, exaggerated dorsal and ventral fins, and truncated tail. In 

 like manner, and using precisely the same co-ordinate networks, it appears to me 

 possible to show the relations, almost bone for bone, of the skeletons of the two 

 fishes ; in other words, to reconstruct the skeleton of the one from our knowledge 

 of the skeleton of the other, under the guidance of the same correspondence as is 

 indicated in their external configuration. 



