in Rectangular Order upon a Medium. 495 



and may be expressed in the usual way as functions P,i(aO of p 5 

 where //, = f/p. Thus 



£3,--*r-p,«. 



In like manner 



and 



dx r' 1 d£ p 1 5 p r * [fM): 



d :i x' o? f .. . 7> . . 



The comparison of terms in (61) thus gives 



A 1 -H=-2B 1 2p-^P 2 -f B 3 V*P«+ . . . 



A 3 =-4B 1 2p- 5 P 4 +.... V. . (62) 



oa or oil— v - a~ ,«.» 



+ 2S ^-^TirrTS,-TO+..., (64) 



In each of the quantities,, such as Sp~~ 3 P 2 , the summation 

 is to be extended to all the points whose coordinates are of 

 the form 



/«, ma, na.^ 



where /, m, n are any set of integers, positive or negative, 

 except 0, 0, 0. If we take a= 1, and denote the corresponding 

 sums by S 2 , S 4 , . . . . , these quantities will be purely nume- 

 rical, and = . 



Ip— 1 ?,, = «-»-%, ....... (63) 



From (52) (62) we now obtain 



Ha 3 2 + v Q a 3 321-v^a 10 

 Br = r^v + ^ 2 ^~"5f+v' 



which with (60) gives the desired result for the conductivity 

 of the medium. 



We now proceed to the calculation of S 2 . We have 



3^-1 2fW-? 2 _ !-4/f\ 

 By the symmetry of a cubical arrangement, it follows that 



2(P/p 5 ) = 2(^ 5 ) = 2(W); 



so that if S were calculated for a space bounded by a cube, 

 it would necessarily vanish. But for our purpose S 2 is to 

 be calculated over the space bounded by £=+co, r)=±v, 



2M2 



