620 
only little under the melting-point of the substance, so that the velo- 
city of crystallisation increases with falling temperature. Then the 
velocity of the boundary surface must be smaller in the axis of the 
tube than at the periphery, i.e. this surface becomes concave towards 
the liquid. The form of the surface can, however, not remain 
unchanged during the increase; as the velocity in normal direction 
is smallest in the axis of the cylinder, and increases towards the 
outside, the curvature will always increase, as is easy to understand, 
and at last a hollow may even arise, which is shut off, and is then 
filled up. At the same time the more rapid growth has proceeded 
at the periphery, and the same thing is repeated. The growth will 
further not be symmetrical round the axis. When through a slight 
disturbance the substance grows somewhat more rapidly at a point 
of the circumference than at the other points, the surface gets here 
further from the places where the crystallisation takes chiefly place, 
i.e. at points where the temperature is lower and the rate of solidi- 
fication, therefore, greater. Consequently the growth in the considered 
point takes place still more rapidly. Hence the condition is unstable. 
A small accidental disturbance will have great influence on the form 
of the boundary surface, hence on the process of the solidification. 
In this case the solidification is a very irregular phenomenon, and 
a theoretical treatment of the problem proposed on p. 619 is 
impossible. 
This is, however, entirely different when the temperature of the 
surroundings, hence that of the tube, is chosen lower, so that the 
velocity of solidification becomes smaller with decreasing temperature, 
Then the normal velocity is greatest in the axis of the cylinder 
where the highest temperature prevails. The surface of the solid phase 
becomes, therefore, convex towards the liquid. This convex surface 
now begins to move parallel to the axis, and in this it assumes a 
very definite form. The normal velocity during this displacement is 
greatest in the axis, and decreases towards the periphery. This decrease 
must be such that in every point the velocity v has the value that 
according to (2c) corresponds to the temperature 6 prevailing there. 
‘here can, and will, arise a condition in which the boundary surface 
moves uniformly and with constant form parallel to the axis. Every 
disturbance in this condition will disappear again of its own accord. 
It is also easy to convince oneself that everything around the axis 
of the tube must be symmetrical. If this is not the case at a moment, 
the growth and conduction of heat takes place in such a way that 
the symmetry is restored. 
Though in this way one can see that the differential equation (1) 
