292 PROCEEDINGS OP THE AMERICAN ACADEMY. 



The Calculations. 



If the deflections obtained from the junctions at different points on 

 the disk were all equal, or nearly so, the calculation of thermal con- 

 ductivity from the observations would be exceedingly simple. Un- 

 equal as these deflections are, it would still be allowable to take their 

 mean as representing the mean difference of temperature between the 

 upper and lower surfaces of the steel disk, if all junctions represented 

 equal areas of the disk ; but this is not the case. The process of find- 

 ing what the mean deflection would be, if all parts of the disk were 

 equally represented, is indicated by the following formula, in which 

 A stands for the mean deflection desired ; 8 for the deflection given by 

 a particular pair of junctions ; cr for the area of cross-section repre- 

 sented by this pair of junctions *, S for the area of one face of the 

 disk : — 



A = — — = — 7^ . 



If there were many pairs of junctions, but only one pair at a given 

 distance, r, from the axis of the disk, each pair would represent a cer- 

 tain ring-shaped portion of the disk, the area of which at either sur- 

 face is 2 7rr X dr. Writing this expression in place of cr, and calling 

 the radius of the disk E, we get 



-^ I 8 X r X dr. 



"We can approximate to this integration by means of a graphical pro- 

 cess, making use of the data given by the observations actually made. 

 From the series of observations given on page 290, we get the follow- 

 ing values ; — 



r 8 



2.88 



2 cm. 3.54 



3.5 " 4.73 



4.5 " 5.33 [= i (5.40 + 5.25)] 



The observed value of S for r = is to be increased a little to make 

 allowance for the fact that the resistance of the circuit, including 

 junctions 13 and 13', was somewhat greater than the resistance of the 



