484 Lord Rayleigh on the Influence of Obstacles 



the shape of the rectangle. The simplest case, which suffices 

 for our purpose, is when we suppose the rectangular boundary 

 to he extended infinitely more parallel to a than parallel to /?. 

 It is then evident that the periodic difference Y x may be 

 reckoned as due entirely to H^, and equated to Ha. For the 

 difference due to the sources upon the axes will be equivalent 

 to the addition of one extra column at + so , and the removal 

 of one at — co , and in the case supposed such a transference is 

 immaterial*. Thus 



V!=H« (7) 



simply, and it remains to connect H with Bj. 



This we may do by equating two forms of the expression 

 for the potential at a point x, y near P. The part of the 

 potential due to H# and to the multiple sources Q (P not 

 included) is 



A o + A 1 rcos0 + A 3 r 3 cos30 + ; 



or, if we subtract H#, we may say that the potential at #, y 

 due to the multiple sources at Q is the real part of 



Ao+(A 1 -H)(^ + ^)4A 3 (^ + ^) 3 H-A 5 (^ + iz/) 5 + ... . (8) 



But if os\ ?/ are the coordinates of the same point when re- 

 ferred to the centre of one of the Q's, the same potential may 

 be expressed by 



2{B 1 (y + « / ')- , + B 3 (*' + «/)- 3 + ....}, . . (9) 



the summation being extended over all the Q's. If £ , rj be 

 the coordinates of a Q referred to P, 



#'=.*-?, y'=y-v; 



so that 



B>' + «y) _ " = B„ (« + iy- Z-i v ) "". 



Since (8) is the expansion of (9) in rising powers of 

 (x-\-iy), we obtain, equating term to term, 



-1.2.3A 3 =1.2.3B 1 S4 + 3.4.5B 3 ^ 6 + ... 

 -1.2.3.4.5A 5 =1.2.3.4.5B 1 S 6 + 3.4.5.6.7B 5 2 8 + . 

 and so on, where 



2 2 „=2(£+^r", en) 



the summation extending over all the Q's. 



* It would be otherwise if the infinite rectangle were supposed to be 

 of another shape, e. g. to ha square. 



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