130 ROYAL SOCIETY OF CANADA 
in the coefficients of expansion of the ice and of the material of which 
the tube is formed has to be taken into account. 
Suppose that the tube is of glass, the coefficient of this material 
being many times smaller than that of ice. When the first layers 
of ice are formed, and the freezing mixture is moved up along the 
tube, both the ice and that part of the tube containing the ice become 
colder, and contract; the ice, however, contracting much more than 
the glass. Thus the ice tends to occupy a smaller fraction of the 
space inside the tube than it would do at zero. This tendency is 
resisted by the rigidity of the mass of ice and its adhesion to the walls 
of the tube, the forces due to the contraction having a shearing effect 
at the juncture of the ice and glass. While the ice is in this con- 
tracted state with reference to the glass vessel, fresh layers of ice 
are continually forming further up the tube, thus binding the ice 
which was previously formed, into its cramped position. This action 
will continue up the tube as long as fresh ice is forming. When this 
process is completed and the apparatus is placed in a zero bath, the 
ice which was first formed tends to expand, this expansion being 
resisted by the superimposed layers. These expansive stresses result 
in a force of compression, longitudinal to the tube, and so tend to 
increase the density of the ice. 
It must be noted that the reasoning in the last case is based 
on the assumption that the end of the extending ice column is flat, 
or approximately so. Should this become concave, as shown by the 
dotted line in fig. 3, the action in the case of the interior formation 
would come in, and under favourable conditions might completely 
offset the effect due to the longitudinal strains in the ice column. The 
concavity of the top of the extending ice formation would be caused 
either by advancing the freezing mixture rapidly, or by using a tube 
of a material of high thermal conductivity. 
Also, in this third formation, the effect due to the shortening of 
the column of ice will be directly proportional to the ratio of the 
coefficient of expansion of ice to that of the material of which the 
tube is formed. Hence, if the latter coefficient were greater than 
that of ice, the column will apparently undergo a longitudinal expan- 
sion during the process of ice formation further up the tube, and so 
the effect of the stresses when the apparatus is placed in a zero bath 
will be to cause the ice to expand and therefore assume a lower density. 
As far as the quantitative aspect of the matter is concerned, little 
can be said, as these actions depend on so many conditions, such as 
diameter of vessels, thermal conductivities, coefficients of expansion of 
materials, etc. | However, these effects should be much more marked 
