530 Prof. A. Macalister on the Mode of Growth of [June 16 



Buccinum subulatum, and describes 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 natu- 

 ralists, as, with the exception of a notice in Professor Goodsir's lecture 

 "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 observations, 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 sur- 

 prised by finding that, in well-formed shells, the ratios of the successive 

 whorls have been specifically constant. In making these measurements, the 

 points to be determined are three, viz. : — 1st, the ratio of elongation of the 

 radius vector of the spiral (A) ; 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 = 0 and 



2nd, those in which n = 0 and m=k t 



3rd, those in which n = m, 



4th, those in which n>m, 



5th, those in which /?<?/?. 

 The cases of discoid shells in which n = 0 are two, the first and second 

 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 producing figure. The result- 

 ing form of this case (which may be formulated thus, k > rri) is an open 

 spiral, as in the fossil Gasteropodous genus Eccyliojnphalus, or the Cepha- 

 lopodous genera Gyroceras, Nautiloceras, and Spirula. The common 

 species of this last genus gives the following measurements : — 



SpiruJa prototypus, m = 2'6, Ar=3'3, n = 0. Generating figure, a circle. 



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

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



* Goodsir's 1 Anatomical Memoirs,' vol. ii. p. 209. 



t 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 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 mea- 

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



