MACALISTER — ON THE GROWTH OF TURBINATED SHELLS. 21 



It will be noted that all these spirals are true logarithmic curves ; 

 and 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 progression being, in discoid shells of the second group m = h, 

 equal to the coefficient of linear increase of the generating figure. To 

 verify the coefficients deduced from the numbers obtained by measure- 

 ment, I have used the method given by the Rev. Canon Moseley, which 

 depends upon a well-ascertained property of the logarithmic spiral, and 

 if fi be taken to represent the ratio of the sum of the lengths of an even 

 number (m) of the whorls to the lengths of half that number, then h = 



2 

 (^- 1) — - Applying this formula to the cases given below, I have in 



the majority of cases obtained results which confirm the ratios of the 

 series of measurements otherwise obtained. 



The second case of discoid shells, in which m — lc and n = 0, is by 

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

 the few exceptions noticed above. The case m > h is one which cannot 

 occur, as then the outer whorl must necessarily crush the inner, and 

 then the generating 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 translations on the axis is not easily measured, and vir- 

 tually amounted to nothing. 



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



least six specimens of each species. These measurements are in decimal parts of an 

 English inch, and were made with a finely-pointed pair of compasses and a diagonal scale, 

 the eye being in some cases aided by a magnifying-glass. Some specimens were measured 

 by means of sections made in a plane perpendicular to the axis. 



