Equipotential Curves and Surfaces, fyc. 15 



electrical image of the other with regard to these circles. This has also 

 been proved to be the case by the experiments (see Plate 2. fig. 2). 



If in an infinite plane there be any number of electrodes, the equi- 

 pofcential curves and lines of flow may be drawn by the method explained 

 by Professor Maxwell *, by drawing circles of equal differences of poten- 

 tial around each charge, and joining their point of intersection in such a 

 way that for all the points taken the sum of the potentials is constant. 



The potential at any point of the plane is + A log r + B log r'-E 

 log r — F log r^&c, where r, r', r v r^ are the distances of the point 

 from the several electrodes. 



In the case where the currents are equal at the several points the 



potential is C-f A log) ? ,\,' ' ' )• Hence for an equipotential curve 



vi r i r i ' ' '/ 



we must have 



r r' r" . . . = c r x r/ r" . . ., 

 or log r+log ?*' + log r" — log r 1 — log r x ' —log r x "=log c. 



By differentiating and finding the curves which cut these at right 

 angles we shall find, as on p. 13, that the equation to a line of flow is 

 of the form 



2 j tan" 1 /fcAVtan- 1 fUl^j\ } = constant, 



or S (0 — 0^ = constant. 



"When currents flow into and out of a plane conducting-sheet at several 

 points of it, the potential at any point of the sheet is 



C + A log r+B log r'-E log r 1 — F log *•/ + &<;., 

 where A, B, &c. are proportional to the quantity of current flowing into 

 the sheet, and E, E, &c. to the quantity flowing out at the several bat- 

 tery-electrodes. In the case of battery-electrodes where there is no loss 

 of current from the conducting-sheet, these currents may always be 

 divided up into a series of pairs of equal and opposite currents, so that 

 there will be an equal number of positive and negative terms. If, then, 



tained in this paper which have been previously worked out experimen- 

 tally. The case worked out by M. Quincke is shown to be the same as that 

 (Plate 2. fig. 3) of one positive electrode at the centre and four negative 

 electrodes at the four corners of a square in the centre of a sheet of tinfoil 

 which is so large compared with the size of the square, that it may be 

 considered of unlimited extent. Some other cases of limited sheets, 

 with other arrangements of electrodes so as to give the same curves, 

 have also been considered in this paper. Since this paper was read before 

 the Royal Society, Prof. Gr. C. Poster has given, in the ' Philosophical 

 Magazine' for May and June 1875, an account of the work which had been 

 previously done on this subject, in which he draws attention to an inter- 

 esting paper on the theory of lines of flow in a plane, by Prof. W. R. 

 Smith, published in the Proceedings of the Roval Society of Edinburgh 

 for 1869-70.] 



* Treatise on Electricity and Magnetism, vol. i. p. 148. 



