MACALISTER — ON THE GEOWTH OF TUEBINATED 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 rector 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 /A 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 = k 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. 



