46 Problems connected with the Flow of Electricity in a Plane. 



same time the inductive capacity of a dielectric medium ex- 

 posed to conductors at different potentials which correspond to 



the poles in the dynamic problem. 



Resistance of some squares and right-angled triangles. 

 § 29. The forms assumed by the resistance-expressions 

 (§8) for the case of a square plate with the poles in certain 

 definite positions may be recorded, together with the products 

 which lead to them. The resistance seems always to be of the 

 form 



,-r, m + n , /AB 



2*\ 



(21) 



w 



where the poles A and B are on angles — and — respectively, 

 1 ° m n 



and where a takes different values in different cases, and is the 



only thing which varies. K still stands for the first complete 



Fig. 10. 



1 Fig. 8. 



elliptic integral with modulus —=-. 



For the case represented in fig. 8 the value 

 of a is zero, because the resistance of such a 

 square is just half that of the right-angled tri- 

 angle (17) § 23. 



When the poles are as in fig. 9, a= \ ; the 

 product in (/3) which leads to the result con- 

 sists of terms coth 2 -~- -, where x takes the va- 

 lues 2, 4, 6, ... . 



The case of fig. 10 gives us a. — J, the pro- 



(m x\ 

 coth 2 -^-), with x successively 



1, 2, 3, 4, &c. 



The case of fig. 11, being unsymmetrical in 

 sources and sinks, requires the complete ex- 

 pression (a) § 8 ; and, because the pole at the 

 corner of the square has to be four times as 

 strong as the other in the infinite sheet, the 

 two factors inside the brackets of (a) occur to 

 different powers ; in other words, calling the 

 factors Pj. and P 2 , we shall have 



Q 5 = PiP^. 

 Now P x , the product referring to the images of the pole at the 

 centre, is the old one whose value is ^- (§ 23). But the pro- 



