4-4 Mr. 0. J. Lodge on some Problems connected 



Bides of the boundaries (§ 4) ; for, in order to arrange poles in 



an unlimited shoot so as to be symmetrical with respect to all 

 three sides of the triangle, it is necessary that one and the same 

 point shall be at the same time both a source and a sink Dr. 

 Henrici suggests that this is possible if the plane consists of 

 two leaves, and recommends in general, whenever both sides of 

 a boundary have to be silvered and some of the images are real 

 (that is, occur in the given plate itself), that these real images 

 be put in another leaf of the plane. It seems just possible 

 that some such contrivance might enable the image-method to 

 be applied to polygons whose angles are not integral submul- 

 tiples of 7T. 



Empirical formula for the resistance of a general isosceles tri- 

 angle. Poles on the equal angles. 

 § 27. Consider a regular polygon of n sides, with one electrode 



(radius p), A, fig. 7, at its centre, 

 and with its entire periphery main- 

 tained at one potential. The re- 

 sistance R« of such a polygon to 

 radial flow is evidently something 

 between that of its inscribed and 

 that of its circumscribing circle ; 

 in other words, 



.log— <R„< 



Fig. 7. 



1 



1 



log 



AC 



2ttk8 p n 27tk8 p 

 Now the resistance R of the isosceles triangle ABC (fig. 7) 

 equals twice the resistance of the triangle AOC, which again 

 equals 2n times the resistance of the polygon. Hence, writing 



AC = AO sec CAO = 1 AB sec -, 



2n . AB ^ 2n . /AB ir\ 

 ^log— <R<^log(^sec-); 



or, calling the angle CAO 0, we may write the resistance of 

 any isosceles triangle with equal angles #, 



R= 



2 /AB 



(*))' 



mr (19) 



where f(0) is something between 2 cos 6 and 2. Hence the 

 limit of Q when the angles of the triangle vanish is J. More- 

 over when the angles are 30°, f(&) must lie between 2 and V3 ; 

 it cannot, therefore, be 1 as the product of images (§ 26) would 

 lead us to believe. 



Looking at the few values of /(#) which are known, one 



