440 Mr. C. H. Lees on the Law of Cooling, and its hearing 



As the term in (^— J is often neglected in the mathematical 



treatment of conductivity*, it is interesting to compare its 

 value, as deduced from experiment, with the first term of the 

 above equation. 



Taking Mitchell's figures for his iron bar, we deduce 



Value of Value of 



1st term. 2nd term. 



At 50° C. excess. -Oil (5'5) = -06 '00001 (2500) = '025 

 „ 100 „ . -011 (13) =-14 -00001 (12,000) =-12 



From which it is evident that the neglect of the second term 

 will seriously affect the results, unless k' is very small, i. e. the 

 conductivity nearly independent of the temperature. In the 

 above case the conductivity has been taken as changing 1 per 

 cent, in 100° C, which, according to the experiments of 

 Forbes f, Kirchhoff and HausemannJ, Lorenz§, and others, 

 is by no means an extreme case. 



Those determinations of conductivity which involve the 

 assumptions k' = 0, n=l, need not be considered, as the results 

 derived from them can only be rough approximations. Inte- 

 grating (4) with respect to x. between the limits x x and x 2 , 

 we have 



f WM/ ; " ; (6) 



or the difference between the amount of heat flowing along 

 the bar at points x^ x 2 is the amount of heat lost from the 

 sides between the two points. 



If there be no " source or sink " along the bar except at 



the origin, and the bar be long enough to make ^— =0 at its 

 end, we have ® x 



qkv hx = ) P^dX) ( 7 ) 



which determines Jc v from observations of v along the bar. 



* The effect of this will be to raise the value of the conductivity as 

 deduced from experiment. 



t Trans. K. S. E. xxiii., xxiv. 



% Wied. Ann. ix., xiii. Kirchhoff and Hausemann neglect, however, 

 the second term in equation (5). 



§ Ibid. xiii. 



