ivith the Flow of Electricity in a Plane, 383 



But we know from Trigonometry that 



. it . Stt 

 Sm 2^ Sm ^ 



. n — j. 

 ( Sm -2n' 7r » 



) . n-2 



VSm-^ 7T 



and that 



Sin -£ 7T 



An 



,-2?r 



2lT . 47T 



fsm-^ — 2 



l In ~ — 



l . w— 1 



V- Sin — ~ 7T 



sm -zr— sm 77— . • . -\ =>l n ~ l s/n 



2n 2n J . n -l 



Sm-7^ 7T 



An 



the upper termination to be taken when n is even, the lower 

 when n is odd. Hence the above product equals 



2rl 



p n } 

 and R = JLi ^ or== J_i _?i , 4) 



if one writes s for the arc subtending of the circle whose 

 radius is r. 



Notice here the close resemblance to (1), becoming identity 

 when r—co , and also the sort of circular analogy to Wallis's 



form of 5- which one gets by taking arcual instead of linear 

 A 



distances in fig. 2 for the product of (/3). 



The circle on which all the poles lie is a flow-line by sym- 

 metry ; and hence the resistance of the sector A B is twice 

 that of the wedge. 



General resistance of a "Wedge" or two-sided polygon, 



poles any where. 



§ 12. If we obtain an expression for the case of two poles A 



and B, it will be little more than a matter of writing to extend 



it to any number. Let X and Y be the two sides of the 



wedge, the angle Y X being 6= — . Call the distance A, 



o\ ; and the angle A X, a (fig. 3) ; and let (r 2 , /3) denote the 

 point B. The images of A will lie in - -pig. 3. 

 pairs on a circle of radius r l3 those of 

 B on a circle of radius r 2 ; and to find 

 the value of the products which occur 

 in the general expression (a) we may 

 make use of Cotes's (or Demoivre's) 

 property of the circle, which says 

 that, if m points equisect the circum- 

 ference of a circle of radius r u the 

 product of their distances from an- 



