MORPHOLOGY AND MATHEMATICS. 



877 



co-ordinates will require greater deformation, and the resultant figure will usually 

 be somewhat less accurate. In fig. 26 I show a network, to which, if we transfer 

 our diagram of Harpinia or of Stegocephalus, we shall obtain a tolerable representa- 

 tion of the aberrant genus Hyperia, with its narrow abdomen, its reduced pleural 

 lappets, its great eyes, and its inflated head. 



The Hydroid Zoophytes constitute a "polymorphic" group, within which a vast 

 number of species have already been distinguished ; and the labours of the systematic 

 naturalist are constantly adding to the number. The specific distinctions are for the 

 most part based, not upon characters directly presented by the living animal, but 

 upon the form, size, and arrangement of the little cups, or " calycles," secreted and 

 inhabited by the little individual polypes which compose the compound organism. 

 The variations, which are apparently infinite, of these conformations are easily 



a. b. c. 



Fig. 27. — a, Campanularia maeruscyphus, Allm. ; b. Gonothyrsea hyalina, Hincks ; c, Clytia Johnstoni, Alder. 



seen to be a question of relative magnitudes, and are capable of complete expres- 

 sion, sometimes by very simple, sometimes by somewhat more complex, co-ordinate 

 networks. 



For instance, the varying shapes of the simple wineglass-shaped cups of the 

 Campanularidse are at once sufficiently represented and compared by means of simple 

 Cartesian co-ordinates (fig. 27). In the two allied families of Plumulariidse and 

 Aglaopheniidse the calycles are set unilaterally upon a jointed stem, and small cup- 

 like structures (holding rudimentary polypes) are associated with the large calycles 

 in definite number and position. These small calyculi are variable in number, but 

 in the great majority of cases they accompany the large calycle in groups of three 

 — two standing by its upper border, and one, which is especially variable in form and 

 magnitude, lying at its base. The stem is liable to flexure and, in a high degree, 

 to extension or compression ; and these variations extend, often on an exaggerated 

 scale, to the related calycles. As a result we find that we can draw various systems 

 of curved or sinuous co-ordinates, which express, all but completely, the configura- 

 tion of the various hydroids which we inscribe therein (fig. 28). The comparative 

 smoothness or denticulation of the margin of the calycle, and the number of its 

 denticles, may constitute an independent variation, and require separate description. 



