1898.] Mr Wilson, On Velocity of Solidification. 33 



If a quantity of heat a- per unit area is instantaneously 

 generated over an infinite plane and left to diffuse, then after a 

 time t the temperature at any point distant x from the plane is 

 given by 



—a; 2 

 I ae lki 



& — 0i = o-j==> 

 * Wirkt 



where k = — the diffusivity. If all the heat goes to one side of 



the plane this result must of course be doubled. In any element 

 of time dt the heat produced on the surface of the solid is 



T? dX J* 



sJ? -j- dt per unit area. 



This will raise the temperature of any point in the liquid at a 

 subsequent time t' by 



-{x-Xf 



sF-T-dt 



dt */irk(t-t)' 



Hence if 0, is the original temperature of the liquid the rise in 

 temperature at any point in the liquid at a time t' is 



-(as-Xja 



6-0,= sF~ f =dt. 

 h dt ^Trkit'-t) 



Differentiating this with respect to x and substituting in 



■ndX jj.dd 



dt dx' 



where x = X', gives 



-(.T'-X) 2 



„dX t* „dX /V/ vx € W-Q ,. 



dt (t = f) Jo <&. ^^{if-tynr 



which is an equation connecting X and if. When t'= 0, this gives 



-sF^ = (oo )dt. 

 dt 



Hence -sF-^ - <|f = Q) = oo . 



7 y 



But -T- = C(6 -6), where is temperature of the solid, 

 etc 



VOL. X. PT. I. 3 



