XI] OF THE MOLLUSCAN SHELL 527 



In the complete mathematical formula (such as 1 have not 

 ventured to set forth*) for any given turbinate shell, we should 

 have, accordingly, to include factors for at least the following 

 elements: (1) for the specific form of the section of the tube, 

 which we have called the generating curve; (2) for the specific 

 rate of growth of this generating curve ; (3) for its specific rate 

 of angular rotation about the pole, perpendicular to the axis ; 

 (4) in turbinate (as opposed to nautiloid) shells, for its rate of 

 shear, or screw- translation parallel to the axis. There are also 

 other factors of which we should have to take account, and which 

 would help to make our whole expression a very complicated one. 

 We should find, for instance, (5) that in very many cases our 

 generating curve was not a plane curve, but a sinuous curve in 

 three dimensions ; and we should also have to take account 

 (6) of the inclination of the plane of this generating curve to the 

 axis, a factor which will have a very important influence on the 

 form and appearance of the shell. For instance in Haliotis it is 

 obvious that the generating curve lies in a plane very oblique to 

 the axis of the shell. Lastly, we at once perceive that the ratios 

 which happen to exist between these various factors, the ratio 

 for instance between the growth-factor and the rate of angular 

 revolution, will give us endless possibilities of permutation of 

 form. For instance (7) with a given velocity of vectorial rotation, 

 a certain rate of growth in the generating curve will give us a 

 spiral shell of which each successive whorl will just touch its 

 predecessor and no more ; with a slower growth-factor, the whorls 

 will stand asunder, as in a ram's horn; with a quicker growth- 

 factor, each whorl will cut or intersect its predecessor, as in an 

 Ammonite or the majority of gastropods, and so on (cf. p. 541). 



In like manner (8) the ratio between the growth-factor and 

 the rate of screw-translation parallel to the axis will determine 

 the apical angle of the resulting conical structure : will give us 

 the difference, for example, between the sharp, pointed cone of 

 Turritella, the less acute one of Fusus or Buccinum, and the 



* The equation to the surface of a turbinate shell is discussed by Moseley 

 [Phil. Trans, torn. cit. p. 370), both in terms of polar coordinates and of the rect- 

 angular coordinates .r. y, z. A more elegant representation can be given in vector 

 notation, by the method of quaternions. 



