99 
This remark seems to mean that passage can be made 
from one state of maximum density to another without 
passing through a state of minimum or maximinimum 
density which has just been shoAvn to be impossible ; indeed, 
if it were not so, the density must be maximum in all the 
intermediate states during the sliding. 
If it is intended to imply that by sliding one layer at a 
time, the change might be made with comparatively slight 
dilation : this is true, but then this is incompatable with a 
uniform strain at the boundaries, and hence lies outside the 
question considered. 
At the top of page 38 it is stated that, “ If a plane layer 
is given in a state of minimum density, there are three sets 
of plane layers in minimum, and three in maximum density.” 
This is not very intelligible, unless it be understood that 
the group is in maximum density. Then it may be under- 
stood, but only by attributing a meaning to the term density, 
which tends to confuse the whole subject. 
A layer of spheres having their centres in a plane can 
have no definable mean density, unless the space occupied 
is defined : that is, unless the upper and lower surfaces are 
defined. From Mr. Gwyther’s statement it appears that he 
considers the bounding surfaces of a plane layer as tangent 
planes to the spheres, but as they occur in the group these 
layers interlock, and hence, according to this definition of 
the density of a layer, two adjacent layers would in part 
occupy the same space. 
The only consistent definition of the density of a layer, 
as it occurs in the group, is the number of spheres in a 
given area of the layer divided by the product of the area. 
