Mr maxwell, ON FARADAY'S LINES OF FORCE. 35 



(14) If the surfaces of equal pressure are known, the direction and magnitude of 

 the velocity of the fluid at any point may be found, after which the complete system of 

 unit tubes may be constructed, and the beginnings and endings of these tubes ascertained 

 and marked out as the sources whence the fluid is derived, and the sinks where it disappears. 

 In order to prove the converse of this, that if the distribution of sources be given, the 

 pressure at every point may be found, we must lay down certain preliminary propositions. 



(15) If we know the pressures at every point in the fluid in two different cases, and 

 if we take a third case in which the pressure at any point is the sum of the pressures at 

 corresponding points in the two former cases, then the velocity at any point in the third 

 case is the resultant of the velocities in the other two, and the distribution of sources is 

 that due to the simple superposition of the sources in the two former cases. 



For the velocity in any direction is proportional to the rate of decrease of the pressure 

 in that direction ; so that if two sj-^stems of pressures be added together, since the rate 

 of decrease of pressure along any line will be the sum of the combined rates, the velocity 

 in the new system resolved in the same direction will be the sum of the resolved parts 

 in the two original systems. The velocity in the new system will therefore be the resultant 

 of the velocities at corresponding points in the two former systems. 



It follows from this, by (9), that the quantity of fluid which crosses any fixed surface 

 is, in the new system, the sum of the corresponding quantities in the old ones, and that 

 the sources of the two original systems are simply combined to form the third. 



It is evident that in the system in which the pressure is the difference of pressure 

 in the two given systems the distribution of sources will be got by changing the sign of 

 all the sources in the second system and adding them to those in the first. 



(16) If the pressure at every point of a closed surface be the same and equal to jo, 

 and if there be no sources or sinks within the surface, then there will be no motion of the 

 fluid within the surface, and the pressure within it will be uniform and equal to p. 



For if there be motion of the fluid within the surface there will be tubes of fluid 

 motion, and these tubes must either return into themselves or be terminated either within 

 the surface or at its boundary. Now since the fluid always flows from places of greater 

 pressure to places of less pressure, it cannot flow in a re-entering curve ; since there are 

 no sources or sinks within the surface, the tubes cannot begin or end except on the surface ; 

 and since the pressure at all points of the surface is the same, there can be no motion 

 in tubes having both extremities on the surface. Hence there is no motion within the 

 surface, and therefore no difference of pressure which would cause motion, and since the 

 pressure at the bounding surface is p, the pressure at any point within it is also p. 



(17) If the pressure at every point of a given closed surface be known, and the 

 distribution of sources within the surface be also known, then only one distribution of 

 pressures can exist within the surface. 



For if two different distributions of pressures satisfying these conditions could be found, 

 a third distribution could be formed in which the pressure at any point should be the 



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