1870.] 



Discoid and Turbinated Shells. 



531 



hence the widths of the whorls measured on the radius vector will form a 

 series of numbers in geometrical progression, the common ratio of the pro- 

 gression being, in discoid shells of the second group where m = /c, equal to 

 the coefficient of linear increase of the generating figure. To verify the 

 coefficients deduced from the numbers obtained by measurement, I have 

 used the method given by the Rev. Canon Moseley, which depends upon 

 a well-ascertained property of the logarithmic spiral, that if p. be taken to 



represent the ratio of the sum of the lengths of an even number (in) of the 



2 



whorls to the lengths of half that number, then ^=(^—1)-. Applying 



this formula to the cases given below, I have in the majority of cases ob- 

 tained results which confirm the ratios of the series of measurements 

 otherwise obtained. 



The second case of discoid shells, in which m=k and w=0, is by far the 

 commoner, as to it belong all genera of discoidal mollusks, with the few 

 exceptions noticed above. The case m > k is one which cannot occur, as 

 then the outer whorl must necessarily crush the inner, and then the gene- 

 rating figure could not retain its geometrical identity while enlarging ; 

 hence we find no examples of it in discoid shells. 



I have placed in this second case some instances in which the ratio of 

 slipping or translation on the axis is not easily measured, and virtually 

 amounted to nothing. 



The following Table of examples illustrate case No. 2 : — 



Species. 



n = 0, 

 k—m 



Generating 

 figure. 



Width of whorls in decimals of an inch. 



Haliotis viridis 



Haliotis rugoso-plicata 



Sulculus (HaliotisJ parvus . . , 

 Padollus (Haliotis) excavatus. 



Natica canrena , 



Nautilus pompilius 



Dolium zonatura 



Solaropsis pellis-serpentis 



Planorbis corneus 



Euomphalus pentangulatus. 

 Architectonica magnifica . . . 

 Architectonica trochleare . . . 



Conus betulinus 



Conus literatus 



Conus virgo 



Planorbis, sp 



Ellipse 



9-3 



-{ 



Oval 



Ellipse 



Segment of 



circle 



Segment of 



ellipsoid. . 



Segment of 

 circle 



Rhomboid 

 Triangle . 



0-0/5 



0-05 



0-15 



0-02 



0-03 



0*03 



0*06 



0'025 

 0-2265 



0-U9 

 0-023 



0-02 



0-124 



0-07 



0-046 

 0-02 . 

 0-03 

 0-08 

 0-03 



0-/5 













0-5 













1-5 













0-18 



1-6 











0-28 



*'7 











0-17 













0-25 



ri 











0-075 



0-25 



0-76 









0-68 



2-04 











0-25 



0-525 











0-047 



0-086 



0-17 



0-34 







004 



0-08 



0-172 









0-25 



0-48 











0-12 



0-2 



035 



0-65 







0-075 



0-175 



0*2 



0-325 



0-55 





0-03 



0-05 



0-072 



0-09 



0-12 



0-17 



0-04 



0-05 



0-86 



0*125 



0-176 



0-25 



o-i 



0-105 



0-16 







0-042 



0*053 



0-078 



o-i 



0'15 



0*18 



0-25 



Hitherto we have been examining the formulse for discoid shells ; but by 

 far the greater number of shell-forms are those in which the whorls, 

 instead of remaining in the same plane, slide down on the central axis, 

 thus making a turbinated shell-form. A new principle enters into our 

 calculation here ; for the shape of a turbinated shell depends on the mutual 

 relation of three, and not two constants. These are, first, the form of the 



