Temperatures of a Tldn Rod. 231 



the elementary equations of conduction and emission, viz.: — 



Q=k 



A.t; Q = «(r 8 -r 4 ).A.< 



(16) 



subject to the conditions above specified. The number thus 

 obtained is a constant for a substance of given conductivity 

 with its radiating surface under definite conditions; and by 

 its means is expressed the relationship of the conducting and 

 radiating powers necessary in the present problem. This 

 thermal constant will be called, for convenience of reference 

 in this paper, the thermal length modulus of the substance. 



If we symbolize this length by L, the Fourier result may 

 be stated in the form 



*~ Vex p(-*/Vy) 



(17) 



with the following two restrictions : — (1) the semi-radius of 

 the rod must be small in comparison with the thermal length 

 modulus ; (2) the geometric mean of these two lengths must 

 be small in comparison with the length of the rod. The first 

 restriction is the condition of the approximation from the 

 rigorous solution for the cylinder referred to in equation (9) 

 above. The second is evident if we consider that the practical 

 realization of the infinitely long bar of the problem is that the 

 radiation at the cold end should be a negligible quantity. 



The following table gives approximate values of the thermal 

 length modulus, in centimetres, for various substances. Two 

 values are given in each case, corresponding to the extreme 



Substance. 



Conductivity. 



Thermal Length Modulus. 



Silver 



109 



•98 



•30 



•26 



•15 



•11 



•083 



•042 



•038 



•027 



•018 



•007 

 •005 

 •004 



•002 



•0006 



•0004 



3630 



3270 



1000 



870 



500 



370 



280 



140 



130 



90 



60 



. 



13 



7 

 o 



T 



5450 



4900 



1500 



1300 



750 



550 



415 



210 



190 



135 



90 



35 

 25 

 20 

 10 

 3 

 2 



Copper 



Zinc 



Brass 



Iron 



German silver 



Lead 



j Antimony 



[ Eose's Metal 



! Wood's Metal 



Bismuth 



Granite 



Marble 



Oak 



Glass 



Paraffin 



Sulphur 





