482 Lord Kayleigh on the Influence of Obstacles 



Since conduction parallel to the axes of the cylinders pre- 

 sents nothing special for our consideration, we may limit 

 our attention to conduction parallel to one of the sides (a) of 

 the rectangular structure. In this case lines parallel to a, 



o 



B 



Fig. 1. 



o 



c 



O 



o 



o 



A 



O 



o 



o 



symmetrically situated between the cylinders, such as AD, 

 liC, are lines of flow, and the perpendicular lines AB, CD are 

 equi potential. 



If we take the centre of one of the cylinders P as origin of 

 polar coordinates, the potential external to the cylinder may 

 he expanded in the series 



V = A o + (Axr + B^" 1 ) cos + (A 3 r 3 -f B 3 r~ 3 ) cos 30 + . . ., (1) 



and at points within the cylinder in the series 



y = C o + C 1 rcos0 + C 8 r 3 cos30 + ..., . . (2) 



being measured from the direction of a. The sines of 

 and its multiples are excluded by the symmetry with respect 

 to 6 == 0, and the cosines of the even multiples by the sym- 

 metry with respect to 0=^7r. At the bounding surface, 

 where r—a, we have the conditions 



V=V, vdY'/dr = dY/dr, 



v denoting the conductivity of the material composing the 

 cylinders in terms of that of the remainder reckoned as unity. 

 The application of these conditions to the term in cosn0 

 gives 



B = 



1+v 



(3) 



In the case where the cylinders are perfectly conducting, 

 v=oc . If they are non-conducting, v = 0. 



