Media for a Steady Flax having a Potential. 22 3 



If v 1 and i* 2 are the potentials in the two media respectively, 

 each satisfies the general condition 



d* 2 + by 2 ~ U ' 

 i>! becomes equal to a constant along A'D', and has no flux 

 across D l C; v 2 becomes constant along B'O', and has no flux 

 across A'B' ; i\ and v 2 become identical and the fluxes due to 

 them equal along A'C 



It might be possible to construct two functions which would 

 satisfy these conditions by adding together particular solutious 

 of the above equation of the form v = e a <- x±i *\ in the way done 

 by Riemann"*, but it is easy by a change of coordinates to 

 make the problem depend on the following simple case, the 

 solution of which is known. 



Let AC (fig. 3) be the line of separation of two media of 



Fia-. 3. 



conductivities k 1 and k 2 , and let AD be an equipotential and 

 DC a stream-line in the k x medium. Let k 1 /k 2 have such a 

 value that AB, the stream-line in the second medium passing 

 through A, makes the same angle with AC that AD does. 

 If, then, CB is the equipotential line in the second medium 

 passing through C, the triangle ABC is equal to the triangle 



If z is the vector coordinate of a point in the plane of 

 fig. 3, and z' the vector coordinate of a point in the plane of 

 fig. 2, it is possible by a transformation z' —f [z) to convert 

 the right-angled triangle ADC in the 2-plane into the isosceles 

 right-angled triangle A'D'C in the /-plane f- Similarly, 

 the same transformation converts the triangle ABC in the 

 ^-plane into the triangle A'B'C in the /-plane; and contiguous 

 points on opposite sides of AC in the £-plane become con- 

 tiguous points on opposite sides of A'C in the /-plane. 

 Hence the quadrilateral ABCD converts into the quadrilateral 

 A'B'C'D', and the difference of potential and the amount of 



* Partielle Differentialgleichungen, Abschnittiv., Bewegung- derWarme 

 t Forsyth, ' Theorv of Functions/ p. 543 ; Love, Amer. Journ. of Math. 

 xi. pp. 164 & 168 (1889). 



