Conduction of Heat in Liquids. 



35 



a temperature in the liquid that was sensibly constant at a constant 

 depth except close to the sides of the tub. The absence of liquid on 

 the negative side of the plane x = might appear a radical defect. It 

 is clear, however, that in the supposed infinite liquid ^f(t) will pass 

 into the liquid on each side of this plane, and the existence of the 

 liquid on the one side merely ensures that •§/(£) is the precise 

 amount passing into the liquid on the other side. But the law of 

 diffusion on either side of the plane can depend only on the heat 

 supplied to that plane, and must be independent of the precise 

 mechanism by which the supply is regulated. For our present 

 purpose it is sufficient to know that f(t) is proportional to the rate at 

 which heat passes into the liquid from the dish, which may be de- 

 termined by a double observation as follows. 



The tub being filled with liquid up to the level of the dish, a certain 

 quantity of water heated to a definite temperature is suddenly poured 

 into the dish. By means of a watch, and a delicate thermometer, 

 raised initially to the temperature of the heated water and with it 

 transferred to the dish, the law of cooling of the water is determined. 

 The quantity of heat lost by the dish per unit of time at any required 

 temperature can be easily deduced. If now the dish be placed on a 

 non-conducting material, and the law of cooling be observed when 

 the other circumstances are the same as before, the quantity of heat 

 which leaves the dish per unit time in the first experiment without 

 passing into the tub is at once obtained for the whole range of tem- 

 perature. From these two experiments it is not difficult to calculate 

 the amount of heat passing into the liquid in the tub a.t every 

 instant in that form of the experiment in which the water poured 

 into the dish was left there. When a siphon was employed the 

 capacity for heat of the water left in the dish and the dish itself was 

 so small that the heat subsequently transferred to the liquid was 

 negligible. 



To a clear understanding of the use of (1) some knowledge of 

 the expression t~ h e~ x2 P c l ikt is desirable. This is proportional to the 

 temperature existing at a depth x in an infinite liquid, originally at 

 zero temperature, at a time t subsequent to the application over the 

 entire plane x = of a unit of heat per unit of area. The first and 

 second differential coefficients of the above expression are re- 

 spectively — 



£-5/2 e -jfipc/4kt ( f\ 



\ 4& 2/' 



and f+m { |^-3 t^t + (^ff } . 



Thus the temperature at depth x, counted from the plane x = 0, 

 gradually commences to rise and continues to do so for a time 



D 2 



