with the Flow of Electricity in a Plane. 389 



strip also, it would become a rectangle of breadth s and length 

 c ± + c 3 or c 2 + c 4 ; and by adding another quantity to (11) similar 

 to the above, but with c 3 and c A instead of c x and c 2 (that is, by 

 adding a quantity which approximates to the effect of the new 

 boundary), we should obtain an approximation to the resist- 

 ance of a rectangle, which is a pretty close one if the c are larger 

 than the s. Such an approximate expression, however, has no 

 particular interest. It is easy to write down the products oc- 

 curring in (a) for the general case of a rectangle ; but their 

 evaluation does not appear to be so simple. 



Distribution of potential in a " wedge " containing any 

 number of point poles. 



rrr 



§ 21. All the flow-conditions in a wedge whose angle is - 



are determined completely by writing out the particular form 

 assumed by the general expression (A)', § 8, for this case. 

 The polar coordinates of the sources being (i\, a), (r\, a') . . ., 

 and of the sinks being (r 2j /3), (r r 2 , $0 • • •; the potential at any 

 point (r, 0) is 



*'&*** _ - ___ 



(r 2n —2r n r n 2 co$ n<j>— $ + rf){r 2n — 2^ cos ncft + ff + rf ) /-^n 

 (r* n — 2r n i\ cos ncj> — a + r\ n ) (r 2n — 2r n r\ cos n<j> + a + rf) 



+ similar terms with accented coordinates for every other pair 

 of poles which the sheet may contain. 



When the wedge becomes a "strip " of breadth s, the above 

 expression gives, as the potential of any point (#, y), 



(cOshQ *- COS *=* 7T ) (cosh^^- 2 7T- COS *±^ 7T ) 



\ S S / \ S S J (IQ^ 



COSH - — 7T— ■ COS — 7T ) ( COSh ^ ^ 7T— COS 7T \ 



S S / V S S ) 



+ &C, 



where the sources are (x l9 ?/i), (x\, y\) . . ., the sinks (x 2 , y 2 ), 

 (a/ i9 y' 2 ) . . ., and where one side side of the strip is taken as the 

 axis of y. 



[To be continued.] 



