XIII] THE TIGHT FIT OF THE TEETH 911 



identity of direction in the two horns of a pair is at once accounted 

 for. 



Beautiful as the spiral pattern of the tusk is, it obviously falls 

 short, in regularity and elegance, of what we find, for instance, in 

 a long tapering Terebra or Turritella, or any other spiral gasteropod 

 shell. In the narwhal we have, as we suppose, only a general and 

 never a precise agreement between rate of torsion and rate of growth ; 

 for these two velocities — of translation and rotation — are separate 

 and independent, and their resultant keeps fairly steady but no 

 more. In the snail-shell, on the other hand, actual tissue-growth 

 is the common cause of both longitudinal and torsional displace- 

 ments, and the resultant spiral is very perfect and regular. 



Before we leave the teeth, let us note that their extreme tightness 

 in their sockets is a remarkable thing. A thin "periodontal mem- 

 brane," less than 0-25 mm. thick, fills up the space between tooth and 

 socket ; and this membrane, elastic, homogeneous and incompressible, 

 is analogous to the thin layer of viscous liquid dealt with in modern 

 theories of lubrication. The equilibrium of the system, the tightness 

 of the fit, the displacement of the tooth under given forces, and 

 the conditions of stress and strain in the membrane, are all open to 

 mathematical treatment ; distributions of pressure can be assigned 

 to the tooth, a centre of rotation can be found, a critical load can 

 be approximately determined, and the pressures calculated at various 

 points. If the membrane thickens, the tooth loosens; its freedom 

 of movement or range of displacement varies with the cube of the 

 thickness of the membrane, and is at most exceedingly small*. 



* J. L. Synge, The tightness of the teeth, etc., Phil. Trans. (A), ccxxxi, pp. 435- 

 477, 1933. 



