CONTRIBUTIONS FROM THE JEFFERSON PHYSICAL LABORATORY, 

 HARVARD UNIVERSITY 



THE ELECTROSTATIC FIELD SURROUNDING TWO 

 SPECIAL COLUMNAR ELEMENTS. 



By P. W. Bridgman. 



Presented by B. 0. Peirce November 8, 1905. Received January 12, 1906. 



Maxwell, in his Electricity and Magnetism, gives diagrams showing 

 the equipotential lines and lines of force surrounding certain two dimen- 

 sional distributions of electrostatic charge. He remarks on the value of 

 these diagrams in enabling us to form a rough idea of what will happen 

 when we have a charged conductor of approximately the same shape as 

 any of the equipotential lines of the diagram. In this paper, diagrams 

 are presented of a few simple cases not given by Maxwell. Before 

 giving a description of the diagrams themselves, a short account will be 

 given of the method by which they were drawn. 



It is well known that the potential due to a two dimensional distribu- 

 tion of electricity is continuous, and in empty space satisfies Laplace's 

 equation, 



dx 2 dy 



Within the body of the charge v 2 </> = 4 7rp, and at surface charges the 

 normal derivative jumps by Arja. Conversely, if we are given a con- 

 tinuous <f>, then a distribution is uniquely determined satisfying the rela- 

 tions above. Either the charge determines the potential (except for an 

 additive constant) or the potential determines the distribution. 



In 1861 Newmann discussed the equation v 2 </> = 0- It appears that 

 this is the necessary and sufficient condition that <f> have a conjugate 

 function \p. This new function if/ also satisfies Laplace's equation, and 

 its level lines are orthogonal to the lines of constant cf> ; that is, the lines 

 of constant \j/ are the lines of force of the distribution of which ^> is the 

 potential. It is evident then that if we are given two continuous 

 conjugate functions, we may regard the level curves of either one of 



