492 Lord Rayleigh on the Influence of Obstacles 



while 



x = k ,2 -k 2 , £= */(&&'). . . . (45) 



I have thought it worth while to quote these expres- 

 sions, as they do not seem to be easily accessible ; but I 

 propose to apply them only to the case of square order, K x = K, 

 **=#=$. Thus 



AB = 1/tt, « = 0, /3=1,V2; . . . (46) 



*o=& *i = 0, * 2 =-2/3 5 , 5 3 =0, 54 =-36/3 9 , 

 and 



„ , x A*x 2/0 f, AV AW 1 /JWX 



Hence 



. 0,(.r) *« A 4 .," A 8 ,. 8 



If i^ij ±^2j • • • are the roots of 0! (#)/#=(), we have 



xx— i, xx— 4*, xx— o, xx— T73 ^ ] . 



Now by (40) the roots in question are 7r (m + z'm') , and thus 



7T 4 A , ~ 



1 A 8 



S 2 = tt, S 4 =^A 4 , S 8 = 7Qt 5P • ■ ( 49 ) 



in which 



,2^ , I 8 1 1*.3 8 1 1 8 .3 8 .5 8 1 



A ~ 7T 2 2 ' 2 2 2 .4 2 'i 2 2 .4 2 .6 2 *8 + " ' 



= 1-18034. 



Leaving the two-dimensional problem, I will now pass on 

 to the case of a medium interrupted by spherical obstacles 

 arranged in rectangular order. As before, we may suppose 

 that the side of the rectangle in the direction of flow is a, the 

 two others being /3 and 7. The radius of the sphere is a. 



The course of the investigation runs so nearly parallel to 

 that already given, that it will suffice to indicate some of the 

 steps with brevity. In place of (1) and (2) we have the 

 expansions 



V = A +(A 1 r + B 1 r- 2 ) Y x +.:... 



+ (A n r» + B„r-»- 1 }Y n +..., . . (50) 

 V' = C + GYS>-r . . . +C«Y„r»+ . . ., . (51) 



