ON FARADAY'S LINES OF FORCE. 173 



It is evident that all that has been proved in (14), (15), (16), (17), with 

 respect to the superposition of different distributions of pressure, and there being 

 only one distribution of pressures corresponding to a given distribution of sources, 

 will be true also in the case in which the resistance varies from point to point, 

 and the resistance at the same point is different in different directions. For 

 if we examine the proof we shall find it applicable to such cases as well as to 

 that of a uniform medium. 



(29) We now are prepared to prove certain general propositions which are 

 true in the most general case of a medium whose resistance is different in 

 different directions and varies from point to point. 



We may by the method of (28), when the distribution of pressures is 

 known, construct the surfaces of equal pressure, the tubes of fluid motion, and 

 the sources and sinks. It is evident that since in each cell into which a unit 

 tube is divided by the surfaces of equal pressure unity of fluid passes from 

 pressure p to pressure (p 1) in unit of time, unity of work is done by the 

 fluid in each cell in overcoming resistance. 



The number of cells in each unit tube is determined by the number of 

 surfaces of equal pressure through which it passes. If the pressure at the 

 beginning of the tube be p and at the end p', then the number of cells in 

 it will be pp'. Now if the tube had extended from the 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 S, the sink of & tubes may be written ?, sinks always being 

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

 system will be 2 (Sp). 



(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 over- 

 coming 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 



