V] 



OF UNDULATING MEMBRANES 



433 



evidence) that the Spirochaete membrane does not undulate, and 

 possesses no thickened border or marginal filament (Randfade)*. 

 It forms a "screw-surface," or helicoid, and, though we might 

 think that nothing could well be more curved, yet its mathematical 

 properties are such that it constitutes a "ruled surface" whose 

 mean curvature is everywhere nil. Precisely such a surface, and of 

 exquisite beauty, may be produced by bending a wire upon itself so 

 that part forms an axial rod and part winds spirally round the axis, 

 and then dipping the whole into a soapy solution. 



Fig. 144, Dinenympha gracilis Leidy. 



A pecuUar type is the flattened spiral of Dinenympha '\ , which 

 reminds us of the cylindrical spiral of a Spirillum among the bacteria. 

 Here we have a symmetrical figure, whose two opposite surfaces 

 each constitute a surface of constant mean curvature; it is evidently 



* For a discussion of this obscure lamella, and of the crista which seems to 

 correspond with it in other species, see Doflein, Probleme der Protistenkunde, ii, 

 Die Natur der Spirochaeten, Jena, 1911; see also ChfFord Dobell, Arch.f. Protisten- 

 kunde, 1912. 



t Leidy, Parasites of the termites, Journ. Nat. Sci., Philadelphia, viii, pp. 425- 

 447, 1874-81; cf. Savile-Kent's Infusoria, ii, p. 551. 



