480 Dr. W. F. G. Swann on the Electrical 



The conductivity is obtained by dividing by X, and the 

 specific resistance s is given by 



/3tt uO (\ 4X W/ , 1 



•=VT-SAii 1+ 3r p w)' • ■ < 9 > 



or writing ¥(a) in fall, and substituting for /www 2 its value 

 3mu 2 /4:3i6, we have 



Sir «0 r 4\r j^r /, ^_ 3mt* 2 \ 



-f^r-'j: .-<■«]}, . . u«> 



■where O is written for (3mi* 2 /4a#) 1/2 . 



Sharpness of the change of specific resistance at the 

 critical value. 



In the first place we notice that the exponential factor 

 and the integral vary so rapidly with u as u increases beyond 

 a certain value, that a very sharp change in conductivity 

 with the change in closeness or packing of the groups may 

 be expected. The value of u will depend upon the time of 

 deposit. In the first stages when the groups are very 

 loosely packed, u may be expected to diminish very rapidly 

 .as t increases, but the rate of variation of u with t must 

 tend to zero as t tends to co . It is impossible for us to 

 attempt to deduce the accurate theoretical relation between s 

 and t unless we know the relation between u and t, but we 

 -can readily see that the general characteristics of the curve will 

 be the same as tlaat of tig 2. Suppose, for example, we take 

 such a relation between u and t as u 2 = hjt, where k is a 

 constant. Such a relation would represent qualitatively the 

 variation of u with t referred to above, and the curve repre- 

 senting s plotted against t would be of a similar shape to 

 that representing s plotted against 1/u 2 . The curve in fig. 4 

 represents the curly bracket of equation (10), which we shall 



A. f) 1 



■call 9i, plotted against tj — 2 -> f° r a case wnere V^ = Ta? ^ 



being constant. s 1 of course here represents the ratio of s 

 to the normal value for the same temperature, and we see 

 that the bend in the curve is a very sharp one, and the curve 

 is in fact very similar in its general characteristics to that of 

 fig. 2. If u varied as a smaller inverse power of t than the 

 first, the bend would of course be less sharp, but the nature 



