CONDUCTIVITY. 97 



the law of conductivity is true, and that the loss to the surroundings from 

 any part of the bar is proportional to the excess of temperature above 

 the surroundings. The curve of temperature ab enabled Despretz to 

 verify Fourier's calculation, and at the same time to obtain comparative 

 results for the conductivity in the different bars. The following special 

 case may sorve to show that it is possible to compare conductivities by 

 means of the temperature-curves. 



Suppose that two bars have the same cross-section, and that their sur- 

 faces are similarly treated, so as to lose the same amount of heat per sq. 

 cm. for the same excess of temperature above the surroundings. Let the 

 hot end of each be at the same temperature, but let the temperature-curve 

 in the first case slope n times as quickly as in the second, so that if T, T' 

 (Fig. 67) represent equal temperatures on the two bars, and the first is at 

 a distance OM from the end of the bar, then the other is at a distance 

 ON = ra.OM. Hence, we may divide the 

 bars into corresponding elements at the 

 same temperature, those of the first bar 



having - the length of those of the 

 n 



second. Then the heat lost beyond any 



point of the first bar is of the heat 

 n 



lost beyond the point of the second bar 

 having the same temperature. 



If, now, the conductivities are 7^ and FIG. 67. 



first bar is to the heat conducted across N on the second as & x x slope at M is 

 to& 2 x slope at N. But the total fall of temperature at M in length 1 is equal 

 to the total fall at N in length n, or the slope at M is n times that at N. 

 Then the quantities of heat conducted across M and N are as k^n to & 2 . 



The heat conducted across a section is equal to the heat lost beyond 

 that section, and we have just shown that the ratio of the heats lost beyond 

 corresponding points is as 1 to n. 



Therefore, 1 : n = k^n : k z 



or, & 2 = k-^n 2 . 



Forbes's Experiment. Forbes modified the method so as to make 

 it give absolute results for the conductivity of an iron bar. His 

 apparatus is shown in Fig. 68. 



The end of the bar is immersed in a constant high temperature bath, 

 and after some hours' heating, the steady state is reached, the temperature 

 at various points of the bar being shown by thermometers, as in Despretz' 

 experiment, inserted in small cavities in the bar containing mercury. 



If the temperature falls through T in a small distance d at any point 

 of the bar, then, as we have seen, the quantity of heat conducted across 

 the section of area A in t seconds is 



Q-Ml. 



a 



The thermometers give r ; A is the cross-section of the bar, and if we 

 can find Q, this equation gives k. 



In order to find Q, Forbes made a subsidiary experiment with a 

 second bar of the same material, but of much smaller dimensions, also 



G 



