Manchester Memoirs, Vol. xliv. (1899), N'o. I. 



I. On the Electrical Resistance between opposite sides 

 of a Quadrilateral one Diagonal of which bisects 

 the other at right angles. 



By Charles H. Lees, D.Sc. 



Received and read October i^ih , iSgg. 



If by means of the equation z =f{z) where z and z 

 are the vector coordinates of two points in two planes, the 

 z plane is conformall)- represented on the z plane, an)- 

 two conjugate systems of curves in the z plane transform 

 into two systems which are conjugate in the z plane.* If 

 the curves in the z plane are the stream and equipotential 

 lines respectively for some flux having a potential, the 

 corresponding curves in the z plane are also stream and 

 equipotential lines, and if the points A, B in the z plane 

 transform into A' B' in the z' plane, the difference of 

 potential between A' and B\ and the flux a cross line 

 joining them, are equal respectively to the difference of 

 potential between A and B and the flux across a line 

 joining them. 



Let A BCD be a quadrilateral in the z plane, the 

 diagonal AC bisecting the other diagonal BD at right 

 angles,so that the triangles ABC D 



and ADC are equal. Let AD 

 and BC be two equipotential 

 lines and AB, Z> 6' stream lines. 

 The resistance of the quadri- A-. 

 lateral may be found if the 

 figure can be transformed by 

 an equation of the above form, 



into another in the z plane for ^ plane. B 



which the resistance is known ^4- i- 



* Maxwell, Eleclricity and Magnetism I., p. 265. 



February 2jrd, igoo. 



