XI] OF THE MOLLUSCAN SHELL 783 



frontal plane, outlined by the lip, and sliding along the axis while 

 revolving round it; but the geometer takes a better and a simpler 

 way. For, when of two similar figures in space one is derived from 

 the other by a screw-displacement accompanied by change of scale — 

 as in the case of a big whelk and a little whelk — there is a unique 

 (apical) point which suffers no displacement; and if we choose for 

 our generating curve a sectional figure centred on the apical point 

 and passing through the axis of rotation, the whole development 

 of the surface may be simply described as due to a rotation of this 

 generating figure about the axis (z), together with a change of scale 

 with the point as centre of simihtude. We need not, and now 

 must not, think of a slide or shear as part of the operation; the 

 translation along the axis is merely part and parcel of the magnifica- 

 tion of the new generating curve. It follows that angular rotation 

 in arithmetical progression, combined with change of scale (from 0) 

 in geometrical progression, causes any arbitrary point on the 

 generating curve to trace a path of uniform gradient round a 

 circular cone, or in other words to describe a helico-spiral or gauche 

 equiangular spiral in space. The spiral curve cuts all the straight- 

 fine generators of the cone at the same angle ; and it further follows 

 that the successive increments are, and the whole figure constantly 

 remains, "self -similar"*. 



Apart from the specific form of the generating curve, it is the 

 ratios which happen to exist between the various factors, the ratio 

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

 revolution, which give the endless possibihties - of permutation of 

 form. For example, a certain rate of growth in the generating 

 curve, together with a certain rate of vectorial rotation, 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 



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

 in terms of polar and of rectangular coordinates, and the method of polar co- 

 ordmates is used also by Haton de la Goupilliere; but both accounts are subject 

 to mathematical objection. Dr G. T. Bennett, choosing his generating curve 

 (as described above) in the axial plane from which the vertical angles are measured 

 (the plane ^ =0), would state his equation in cylindrical coordinates, / {za^, ra^) =0: 

 that is to say in terms of z, conjointly with ordinary plane cylindrical coordi- 

 nates. 



