621 
with the boundary conditions (2) perfectly determine the form of the 
boundary surface of the phases on solidification in a tube, this 
determination is attended by great mathematical difficulties. We shall, 
therefore, suppose for simplification that tbe surface of the solid phase 
is a plane at right angles to the axis of the tube’). The constant 
velocity v, with which this plane moves, is determined according to 
(2c) by the temperature 6 at this plane. 
When there shall actually arise a condition in which the boundary 
plane, preserving its shape, moves uniformly, the whole distribution 
of temperature also in solid and liquid phase will have to move. 
with it with this velocity, in other words, the temperature will only 
depend on the distance from the boundary surface. That a solution 
of (1) and (2) with this property actually exists, will now be shown. 
In the solid substance, where the matter is at rest, and the condition 
round the axis is symmetrical, the differential equation (1) assumes 
the form: 
oo; 6 ores = Ld 00, : 
ee rs oe fee | tad ts en (0) 
| } 
in which a, —— and £ is a coordinate, which is measured along 
Cg, 
the axis of the tube in the direction of the velocity v with which 
the boundary surface moves, and 7 the distance from the axis. 
On solidification contraction takes place. In consequence of this the 
liquid moves in a direction opposite to that of the positive &-axis 
with a constant velocity V, which in the densities y, and o, of the 
solid and the liquid phase can be expressed thus: 
V=— Gee v. 
0; 
Accordingly the differential equation holding in the liquid, becomes: 
00, OO! 108 DEE 0,—0, 906, r 
gf Ty EEM RED OTE ® 
_ When the temperature in the solid and the liquid phase is sup- 
posed only to depend on the distances w, resp. a, from the boun- 
dary surface, the differential quotients according to time may be 
expressed in those according to place: 
. PO nahin 06 (ee 
En Det lth de 
nn rn | 
Further: 
*) As in the cases that occur most frequently the velocity v depends only little 
on the temperature, the boundary surface will generally be only little curved. 
40 
Proceedings Royal Acad. Amsterdam. Vol. XXIII. 
