298 



Mr. A. S. Davis on the Vibrations which Heated 



■ where V is half the difference between the initial temperatures 

 of the two pieces of metal, v is their arithmetic mean, k is 

 the conductivity of the metal in terms of the thermal capacity of 

 the unit of bulk. (See Thomson and Tait, ' Natural Philo- 

 sophy/ vol. i. p. 717.) 



If Y be taken in the 

 surface of the metal and 

 X vertically downwards, 

 and if we take A to re- 

 present V, and N P to re- 

 present v — (v — V) (i. e. 

 the excess of the tempe- 

 rature of the metal at the 

 depth ON (=#) at the 

 time t over the initial tem- 

 perature of the metal), the 

 curve A P Q will be very 

 near the axis of X at all 

 depths greater than 4<\/kt, because the value of 



1 C w 

 —j^ I e~ z2 dz is very nearly equal to \ for all values of w> 2. 



Take B = 2\/kt, and complete the rectangle D. 



The expansion which the metal has undergone will be pro- 

 portional to the area A Q ; and this expansion bears the same 

 proportion to the expansion which the metal would undergo if 

 heated uniformly to a temperature v down to a depth B as 

 the area A Q bears to the area of the rectangle D. The 

 ratio which the area A Q bears to D is constant, and by 

 careful measurement is found to be nearly as 6 : 10. (This curve 

 is carefully drawn in Thomson and Tait' s ' Natural Philosophy/ 

 vol. i. p. 719.) 



Let d be the increase of unit length of the metal for an in- 

 crease of one degree in the temperature. 

 through which the surface has risen in time t is 



the integral 



Then the height 



-6xYx2y / ktxd. ...... (2) 



When the hot and cold metals are of the same kind, the con- 

 traction of the hot metal will be equal to the expansion of the 

 cold metal (supposing the dilatability not to vary with the tem- 

 perature) . In this case, then, the mass of hot metal will not be 

 raised by the flow of heat. 



Let us next consider the case in which the hot and cold me- 

 tals are of different kinds. The common temperature of the 

 surfaces in contact will not be, as in the former case, a mean 

 between the initial temperatures of the two metals. 



