Mb maxwell, ON FARADAY'S LINES OF FORCE. 41 



source to a place where the pressure is zero, the number of cells would have been p, and if 

 the tube had come from the sink to zero, the number would have been p', and the true number 

 is the difference of these. 



Therefore if we find the pressure at a source S from which S tubes proceed to be p, Sp 

 is the number of cells due to the source S; but if S" of the tubes terminate in a sink at 

 a pressure p', then we must cut off S'p' cells from the number previously obtained. Now if 

 we denote the source of S tubes by 5*, the sink of S' tubes may be written - ,S^, sinks always 

 being reckoned negative, and the general expression for the number of cells in the system will 

 be 2 iSp). 



(30) The same conclusion may be arrived at by observing that unity of work is done on 

 each cell. Now in each source S, S units of fluid are expelled against a pressure p, so that 

 the work done by the fluid in overcoming resistance is Sp. At each sink in which 

 S' tubes terminate, ^S^ units of fluid sink into nothing under pressure p'; the work done upon 

 the fluid by the pressure is therefore S'p'. The whole work done by the fluid may therefore 

 be expressed by 



TF=2Si)-25'y, 

 or more concisely, considering sinks as negative sources, 



W='2(Sp). 



(31) Let S represent the rate of production of a source in any medium, and let p be 

 the pressure at any given point due to that source. Then if we superpose on this another 

 equal source, every pressure will be doubled, and thus by successive superposition we find that 

 a source nS would produce a pressure np, or more generally the pressure at any point due to 

 a given source varies as the rate of production of the source. This may be expressed by the 

 equation 



p = RS, 



where iZ is a coeflicient depending on the nature of the medium and on the positions of the 



source and the given point. In a uniform medium whose resistance is measured by k, 



kS „ Ar 



p = - — , .-. B = - — , 



4nrr 47rr 



R may be called the coefficient of resistance of the medium between the source and the given 

 point. By combining any number of sources we have generally 



p = ^(RS). 



(32) In a uniform medium the pressure due to a source S 



_k_ S 

 47r r 

 At another source (S^ at a distance r we shall have 



S'p= = Sp, 



if p' be the pressure at S due to S. If therefore there be two systems of sources 2(5') and 

 2(y), and if the pressures due to the first be p and to the second p', then 



2(yp) = 2(V). 



For every term S'p has a term Sp' equal to it. 



Vol. X. Paet I. 6 



