1898.] Mr Wilson, On Velocity of Solidification. 31 



The rate of production of heat at the surface is per unit area 

 vsF, using the same notation as before. The rate at which heat is 

 produced at the surface must be equal to the rate at which it is 

 conducted away from the surface, consequently at any point on 

 the surface 



T, Ty . dd Tr do _ 



vsF+ K p -=- + K s -=— = 0, 

 au p du s 



where u p and u s are the normals to the surface at the point in the 

 liquid and solid respectively, and K p and K s are the conductivities 

 of the liquid and solid respectively. 



These two equations v = G(0 O — 0) and 



-f, T , du Tr do _ 

 vsF + K p -=— + K s -j— = 0, 

 du p du s 



together with the ordinary equations for heat conduction, are 

 sufficient to enable any problem in solidification to be solved. 



The equation v — G (0 O — 0) does not take into account any 

 tendency of the solid to crystallise out in particular crystalline 

 forms. In the case of phenol and other substances the solidifi- 

 cation appears to be determined by the temperature alone, but 

 this does not hold good for all cases. Thus melted sodium thio- 

 sulphate when supercooled solidifies partly in rays like phenol, 

 but in addition to the rays crystals are also sometimes formed 

 which do not grow at a uniform rate like the tips of the rays. 



I shall now consider the case of a long hollow circular cylinder 

 of ice, filled with water, whose external surface is kept at a con- 

 stant temperature 9 1 which is below the melting point of the 

 ice O . Let r x and r 2 be the internal and external radii of the 

 cylinder. Then if 6 is the temperature at the internal surface 

 the amount of heat conducted through the cylinder per unit 

 length in unit time is, if the temperature of the internal surface 

 and r 1 are supposed to vary very slowly, 



log- 2 



When r x and r 2 are very nearly equal, r, will change rapidly, so 

 that the above expression, which assumes that a steady state has 

 been reached, will not be applicable. Also when the cylinder is 

 thick must be very nearly equal to O , so that practically all the 

 heat produced by the solidification is conducted away through the 

 ice, consequently 



