20 NATURAL HISTORY SOCIETY OF DUBLIN. 



scribes many interesting particulars regarding the formation and growth 

 of the operculum in different shells. 



This subject does not seem to have attracted much attention from 

 naturalists, as with the exception of a notice in Professor Goodsir's lec- 

 ture " On the Use of Mathematical Modes of Investigating Organic 

 Forms,"* it is not, to my knowledge, referred to by any writer on 

 zoology. 



While engaged in arranging the large collection of shells in the 

 Museum of the University of Dublin, I was led to make measurements 

 of univalve shells in order to see whether any deduction of zoological 

 importance might be drawn from these valuable geometrical observa- 

 tions, and more especially to determine whether it might be possible to 

 arrive at constant specific numerical parameters in these cases ; and in 

 all instances I have been surprised by finding that, in well-formed 

 shells, the ratios of the successive whorls have been specifically con- 

 stant. In making these measurements, the points to be determined are 

 three, viz.: — 1st, the ratio of elongation of the radius vector of the 

 spiral (Jc) ; 2nd, the degree of linear expansion of the generating figure 

 in the successive whorls (m) ; and 3rd, the degree of translation or 

 slipping of the spiral on the central axis (n). The second of these 

 we may call the discoidal coefficient, and the third the helicoidal 

 coefficient. 



On applying these measurements to univalve shells, we find that 

 the possible combinations are five in number : — 



1st, those in which n = and m < k, 

 2nd, those in which n = and m = Jc, 

 3rd, those in which n = m 

 4th, those in which n > m 

 5th, those in which n < m. 



The cases of discoid shells in which m = are two, the first and se- 

 cond on the list. The first and most uncommon is that in which the 

 amount of elongation of the radius vector in the formation of the 

 successive whorls exceeds the transverse linear increase of the produ- 

 cing figure. The resulting form of this case (which maybe formulated 

 thus, %>m) is an open spiral, as in the fossil Gasteropodous genus 

 Eceyliomphalus, or the Cephalopodoas genera Gyroceras, Nautiloceras, 

 and Spirula. The common species of this last genus gives the following 

 measurements : — 



Spirula prototypus, m = 2-6, Jc = 33, n = 0. Generating figure, a circle. 

 Average width of whorls 0-075 in., 0*2 in.f 



* Goodsir's "Anatomical Memoirs," vol. ii. p. 209. 



f In all the specimens measured and referred to in this paper I have made at least 

 three measurements of each individual, and in the majority of cases I have measured at 



