Metals undergo when in contact with cold Material. 299 



To find what this temperature will be, let T be its excess over 

 the initial temperature of the cold metal, and let T ; be the excess 

 of the initial temperature of the hot metal over the same tempe- 

 rature. Then if d } d ] , p, p' } a } a 1 be respectively the disabilities, 

 specific gravities, and specific heats of the cold and hot metals, 

 it will be seen from what has been already proved that the ex- 

 pansion of the cold metal is to the contraction of the hot metal as 



But if h be the heat which has flowed out of the hot metal into 



the cold metal, the expansion of the cold metal is — ; for the 



pa 



total expansion would be the same however the heat were distri- 

 buted : and it is clear that — would be the expansion if the heat 



h were uniformly distributed over a unit cube of the metal. In 



hd! 

 the same manner the contraction of the hot metal will be -j— .. 



p'a' 



Thus we have 



TdVki: (T.-TWVW: : - : ^ 

 . per pa 



whence 



T-T y pWff , . 



When the two metals are of different kinds, the expansion of the 

 one will not in general be equal to the contraction of the other, 

 and consequently the mass of hot metal will be either raised or 

 lowered. 



Let us now consider the case in which the hot metal is copper 

 and the cold metal lead. In this case 



p = 11-35 p , =8-88 



a= -0314 a'= -095 



d=z 0-000028 d'= 0-000017. 



Taking a decimetre as unit of length, and a second as unit of 

 time, and 1° C. as unit of temperature, 



v/W0327, v 7 A 7 =-0475. 



Substituting, we obtain 



T=-77xT r 



Hence the expansion of the lead is 



1-2 x 0-000028 x -0327 x '77 x T, x n/F 

 = -000000846 xTxv^j 



