388 Mr. 0. J. Lodge on some Problems connected 



Whether this result is otherwise obvious or not, it will serve 

 as an example of how new products might sometimes be evalu- 

 ated from resistance-expressions. 



General resistance of a circular sector. 



§ 18. The resistance of a circular sector containing two 

 poles A and B will be twice the resistance of a wedge with the 

 same angle, but in which, besides the poles A and B, there 

 also exist poles A! and W at the points inverse to A and B with 

 regard to the circle of which the sector is a part ; that is, calling 



R 2 ll 2 



R the radius of the sector, A' = — , 0B' = — . The ex- 



r x r 2 



pression for the resistance will be (5) plus an additional 



expression 



§ 19. When the angle of the sector vanishes by R becoming 

 infinite, we get the case of a strip bounded towards one end, 

 or what may be called an isosceles triangle whose equal angles 



rrr 



are «-. The pole A being at a distance c x from the base and 



a from one of the sides, while B is at a distance b from the 

 same side, c 2 from the base, the above expression reduces to 



(■ L c 1 + c 2 a — b \ / , <?i + c 2 a + b \ 

 COSh 7T — COS 7T ) ( COSH 7T— COS — It 1 

 s s )\ s s / 



2 smh — ^smn — —\ / ( cosh -— cos )\ / \ cosh — cos 



s sVV s s / v \ s s / 



where the c in (11) may be written c 1 — c 2 for symmetry. 



It may be noticed that the additional quantity in (14) vanishes 



when Ci and c 2 are infinite; but becomes the same as (11) 



when Ci and c 2 are made small and equal to ^; and this is just 



what it ought to do. 



§ 20. The quantity which is added to (11) to give (14) may 

 be called the " effect of the boundary " which limits the strip 

 at one end. If we added a bound ary to the other end of the 



