490 Lord Rayleigh on the Influence of Obstacles 



The reason for this will appear more clearly if we consider 

 the nature of the first summation (with respect to mf) m In 

 (19) we have to deal with the sum of (%-\-iy)~ n , where y is 

 for the moment regarded as constant, while x receives the 

 values x=m t . If, instead of being concentrated at equidistant 

 points, the values of x were uniformly distributed, the sum 

 would become 



" +0 ° dx 



I 



(x + iy) n ' 



Now, n being greater than 1, the value of this integral is 

 zero. We see, then, that the finite value of the sum depends 

 entirely upon the discontinuity of its formation, and thus a 

 high degree of convergency when y increases may be ex- 

 pected. 



The same mode of calculation may be applied without 

 difficulty to any particular case of a rectangular arrangement. 

 For example, in (11) 



Z 2 = 2 {ml* + im/3) ~ 2 = a~ 2 S (m ! + imfi/a) ~ 2 . 



If m be given, the summation with respect to m! leads, as 

 before, to 



and thus 



m=oo 



a%=27r 2 2 sin- 2 (irmrP/a) + £tt 2 . . . (36) 



The numerical calculation would now proceed as before, 

 and the final approximate result for the conductivity is given 

 by (16). Since (36) is not symmetrical with respect to a 

 and /3, the conductivity of the medium is different in the two 

 principal directions. 



When /3 = ct, we know that a- 2 2 2 = '7r. And since this does 

 not differ much from ^7r 2 , it follows that the series on the right 

 of (36) contributes but little to the total. The same will be 

 true, even though /3 be not equal to a, provided the ratio of 

 the two quantities be moderate. We may then identify a _2 2 2 

 with 7r, or with ^7r 2 , if we are content with a very rough 

 approximation. 



The question of the values of the sums denoted by X 2 n is 

 intimately connected with the theory of the ^-functions *, 

 inasmuch as the roots of H(w), or 1 (7ru/2~K), are of the form 



2mK+2m'eK'. 



* Cayley's ( Elliptic Functions/ p. 300. The notation is that of Jacobi. 



