ON FARAD AYS LINES OF FORCE. 165 



to be considered.) There are therefore two sets of surfaces which by their inter- 

 section form the system of unit tubes, and the system of surfaces of equal pres- 

 sure cuts both the others at right angles. Let h be the distance between two 

 consecutive surfaces of equal pressure measured along a line of motion, then since 

 the difference of pressures = 1, 



kvh=l, 



which determines the relation of v to h, so that one can be found when the 

 other is known. Let s be the sectional area of a unit tube measured on a 

 surface of equal pressure, then since by the definition of a unit tube 



vs = l, 



we find by the last equation 



s = kh. 



(13) The surfaces of equal pressure cut the unit tubes /into portions whose 

 length is h and section s. These elementary portions of unit tubes will be called 

 unit cells. In each of them unity of volume of fluid passes from a pressure p to 

 a pressure (p 1) in unit of time, and therefore overcomes unity of resistance in 

 that time. The work spent in overcoming resistance is therefore unity in every 

 cell in every unit of time. 



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

 tude 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 end- 

 ings 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 systems of pressures be added 



