in Rectangular Order upon a Medium. 483 



The values of the coefficients A^B^ A 3 ,B 3 . . . are neces- 

 sarily the same for all the cylinders, and each may be regarded 

 as a similar multiple source of potential. The first term A , 

 however, varies from cylinder to cylinder, as we pass up or 

 down the stream. 



Let us now apply Green's theorem, 



J( 



<-<y=» ■ ■ ■ ■ w 



j- 



to the contour of the region between the rectangle ABCD and 

 the cylinder P. Within this region V satisfies Laplace's 

 equation, as also will U, if we assume 



U = #=rcos# (5) 



Over the sides BC, AD, dTJ/dn, dY/dn both vanish. On CD, 

 dY/dn ds represents the total current across the rectangle, 

 which we may denote by C. The value of this part of the 

 integral over CD, AB is thus «C, The value of the remainder 

 of the integral over the same lines is — V^, where Y x is the 

 fall in potential corresponding to one rectangle, as between CD 

 and AB. 



On the circular part of the contour, 



U = a cos 0, dV/dn = — dJJ/dr = — cos ; 



and thus the only terms in (1) which will contribute to the 

 result are those in cos 0. Thus we may write 



Y=(A 1 a-fB 1 a- 1 )cos6>, 

 dY/dn= - (Ai-B^- 2 ) cos 0; 



so that this part of the integral is 27rB v The final result 

 from the application of (4) is thus 



«C~/3V 1 + 2ttB 1 = (6) 



If Bj = 0, we fall back upon the uninterrupted medium of 

 which the conductivity is unity. For the case of the actual 

 medium we require a further relation between Bj and Y v 



The potential V at any point may be regarded as due to 

 external sources at infinity (by which the flow is caused) and 

 to multiple sources situated on the axes of the cylinders. The 

 first part may be denoted by H#. In considering the second 

 it will conduce to clearness if we imagine the (infinite) region 

 occupied by the cylinders to have a rectangular boundary 

 parallel to a. and /3. Even then the maimer in which the infinite 

 system of sources is to be taken into account will depend upon 



