in Rectangular Order upon a Medium. 



497 



approximate value of S 4 by direct summation from the 

 formula 



We may limit ourselves to the consideration of positive and 

 zero values of f, 77, f. Every term for which f, rj, f are 

 finite is repeated in each octant, that is 8 times. If one of 

 the three coordinates vanish, the repetition is fourfold, and 

 if two vanish, twofold. 



The following table contains the result for all points which 

 lie within p 2 =18. This repetition in the case, for example, 

 of p 2 =9 represents two kinds of composition. In the first 



and in the second 



^ = 2 2 + 2 2 + l 2 = 9, 

 / >2 = 32 + 2 + o 2 = 9. 



The success of the approximation is favoured by the fact that 

 F vanishes when integrated over the complete sphere, so 

 that the sum required is only a kind of residue depending 

 upon the discontinuity of the summation. 

 The result is 



S 4 = 3-ll (69) 





P 2 





3-5000 





P 2 





0, 0, 1 



1 



4- 



0, 0, 3 



9 



+ -0144 



0, 1, 1 



2 



- 



•3094 



0, 1, 3 



10 



+ -0243 



1, 1, 1 



3 



— 



•1996 



1, I, 3 



11 



4- -0075 



0, 0, 2 



4 



4- 



•1094 



2, 2, 2 



12 



- 0062 



1 0, 1, 2 



5 



+ 



•0501 



0, 2, 3 



13 



- -0015 



1, 1. 2 



6 



— 



•0397 



1, 2, 3 



14 



- -0095 



0, 2, 2 



8 



— 



•0097 



0, 0, 4 



16 



+ -0034 



1, 2, 2 



9 





0277 



2, 2, 3 

 0, 1, 4 



17 



17 



- 0061 



4- '0085 



The results of our investigation have been expressed for 

 the sake of simplicity in electrical language as the con- 

 ductivity of a compound medium, but they may now be 

 applied to certain problems of vibration. The simplest of 

 these is the problem of wave-motion in a gaseous medium 

 obstructed by rigid and fixed cylinders or spheres. It is 

 assumed that the wave-length is very great in comparison 

 with the period (a ; /3, 7) of the structure. Under theso cir- 



