On the Growth of Discoid and Turbinated Shells. 261 
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 a have been n spe- 
cifically constant. In making these measurements, the points to be 
determined are three, viz. :—Ist, the ratio of elongation of the radius 
vector of the spiral (4); 2nd, the degree of linear expansion of the 
generating figure in the successive M (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 iets coefficient, and the third 
the helicoidal coefficient. 
On applying these measurements to univalve shells, we find that 
the possible combinations are five in number :— 
Ist, those in which n=0 and m—£, 
2nd, those in which 420 and n—£, 
3rd, those in which n=m, 
4th, those in which n>m, 
5th, those iu which næm. 
The cases of se shells in which n-0 are two, the first and 
second on the list. The first and most nncommon is that in which 
the amount of itid of the radius vector in the formation of 
the successive whorls exceeds the transverse linear increase of the 
producing figure. The resulting form of this case (which may be 
formula ted thus, k > m) is an open spiral, as in the fossil Gastero- 
podous genus Kee yliomphalus, or the ^ yerbum genera Gyro- 
ceras, Nautiloceras, and Spirula. The common species of this 
last genus gives the following pierdas :— 
Spirula Spee ypus, m=2°6, k—3:3, n=0. Generating figure, a 
. Average width of whorls 0:075 in., 02 in 
It will be noted that all these spirals are true logarithmie curves ; 
and hence the widths of the whorls m ensured on “the radius vector 
omm 
obtained by measurement, I have used the method given 
Rev. Canon Moseley, which depends upon a well-ascertained property 
of the logarithmic spiral, that if u be taken to represent the ratio of 
the sum of the lengths of an even number (m) o of the whorls to the 
lengths of half that number, then 4—(u— DE Applying this 
formula to the cases given below, I have in the edo of cases ob- 
* In all the specimens measured and referred to in this paper, I have made at 
least three vereri 2 -— individual; and in the y OF case x 
fasured at le. ix of nei de ese measurements are in 
decimal parts at a an Engh į mak, and w ith ] : 
Compasses and a diagonal scale, the e peni in some cases aided by a magnify- 
ing-glass. Some specimens pue nig ed by means of sections made in a plane 
Perpendicular to the axis. 
