Absolute Thermal Conductivity of Crystals fyc. Ill 



and the rest exposed to the atmosphere. Let s be the area of 

 cross section (which need not be circular) and P the perimeter 

 of the rod, the latter being defined as the length of a string 

 wrapped once round the rod if the actual perimeter is any re- 

 entrant curve. The condition to be expressed is that the total 

 gain of heat of any element of the rod by its anterior, posterior, 

 and exterior surfaces is equal to nothing. Taking t as the 

 excess of temperature over that of the air of an element in a 

 position x along the rod, we have the quantity of heat 



, dt entering at its anterior or hotter surface in unit 

 "" s ~dx time, 



+ ks I - — I- -r-: 2 dx ) at its posterior surface, 



— Tht dx at the surface exposed to the air ; 



h being the conductivity of the rod, and h the radiation-coeffi- 

 cient of its surface, i. e. the quantity of heat lost in unit time 

 by unit surface when it is one degree hotter than the air. 

 Putting the sum of these quantities equal to zero, we have 



dh Ph 2 



an equation whose complete integral is 



t=G 1 eP*+C 2 e-P*, 

 or, as I shall prefer to write it, 



t = A cosh px — B sinh px. 



The constants A and Bare determined in terms of p as soon 

 as one knows the temperatures of any two points of the rod ; 

 and the temperature of a third point will determine jo, whence, 

 if h be separately found, k is known. Thus, suppose we know 

 t the temperature of the origin, and t x the temperature of a 

 point at a distance x from it, and also tx the temperature half- 

 way between these points ; then 



A_£ , 



B = t coth px — t x cosech px, 



2. Now suppose the rod to be cut in half and a slice of 

 crystal or any substance interposed : the curve of temperature 

 will have a discontinuity at the junctions ; but if the curve along 

 each rod is observed, it may be possible to calculate it for the 

 crystal. The method which I propose, then, is to cut a cylin- 

 drical piece of the substance to be examined, of length g f with 



