W. F. E. Weldon 
111 
at the apex of the shell, the angle B will describe a spiral which will be 
identical with the peripheral spiral of the shell if the law of change of the side 
AB is the same as that of the peripheral radius vector of the shell; and by 
finding an appropriate law of change for the sides AG and BG, the angle G may 
be made to describe the columellar spiral, while the side BG maintains the two 
spirals at their proper distance apart. We may therefore consider the two funda- 
mental spirals, upon wliich the main characters of the shell depend, as generated 
by a triangle revolving round a long and narrow cone, so that one side always 
touches the cone, and one angle always lies at the apex of the cone, the length of 
each side changing as the triangle revolves. 
In actual sections of shells, such as that drawn in Fig. 1, the peripheral and 
columellar spirals are each cut from 17 to 21 times; so that the elements of 
the fundamental triangle can be determined a great number of times. Further, 
since the section is approximately flat, the interval between any position ABG 
and the next position AB'G' in which the triangle can be measured, corresponds 
to a revolution through 180°. From such a number of data it should be possible 
to determine the law of change of the triangle with considerable accuracy. The 
measurements required are somewhat laborious, and I have not at present 
determined the rate of change of every side of this fundamental triangle, but 
only that of the side AB, the radius vector of the peripheral spiral. Such a 
determination leaves the distance between the columellar and peripheral spirals 
uncertain : it only determines that the radius of the peripheral spiral at any 
point lies somewhere on the arc of such a circle as that shown at FEG, Fig. 1. 
The results of this determination seem to me of some interest, so that I venture 
to publish them before the other measures are completed. A reason for considering 
these measures separately from the others is that they do not seem to be affected 
by a curvature of the upper part of the columella, which is often sensible, and 
sometimes considerable. The shells measured were chosen at random, and include 
some in which the columella is very sensibly bent. 
The chief material used was collected during the summer of last year and 
during the present summer in the great beech-wood known as Der Holm at 
Gremsmiihlen on the Dieck See, a lake on the eastern border of Holstein. In 
this wood Glausilia laminata was extremely abundant, occurring together with 
G. biplicata. Other species also occurred, but none were so common as G. laminata. 
The conspicuous species of Helix were H. lapicida, H. nemoralis, H. arhustorum, 
H. hispida and H. rotimdata, all of which were exceedingly common. The forest 
slopes down to the very edge of tlie lake, and the moist nature of the ground 
where the Glausilia were collected is shown by the frequent presence of species of 
SuGcinea. The soil is the light sandy alluvium so common throughout Holstein, 
and the whole wood is probably less than 150 feet above the sea level. 
The peripheral radii were measured in sections of 100 adult shells from this 
wood. The measures were recorded to O'Ol millimetre, and I believe they are 
fairly trustworthy. They were made with an instrument designed for me by 
