$4 Me maxwell, ON FARADAY'S LINES OF FORCE. 



Any portion of the fluid moving through the resisting medium is directly opposed by a 

 retarding force proportional to its velocity. 



If the velocity be represented by v, then the resistance will be a force equal to kv acting on 

 unit of volume of the fluid in a direction contrary to that of motion. In order, therefore, that 

 the velocity may be kept up, there must be a greater pressure behind any portion of the 

 fluid than there is in front of it, so that the difl^erence of pressures may neutralise the effect of 

 the resistance. Conceive a cubical unit of fluid (which we may make as small as we please, 

 by (5)), and let it move in a direction perpendicular to two of its faces. Then the resistance 

 will be kv, and therefore the diff^erence of pressures on the first and second faces is kv, so that 

 the pressure diminishes in the direction of motion at the rate of kv for every unit of length 

 measured along the line of motion ; so that if we measure a length equal to h units, the dif- 

 ference of pressure at its extremities will be kvh. 



(11) Since the pressure is supposed to vary continuously in the fluid, all the points at 

 which the pressure is equal to a given pressure p will lie on a certain surface which we may 

 call the surface (p) of equal pressure. If a series of these surfaces be constructed in the fluid 

 corresponding to the pressures 0, 1, 2, S &c., then the number of the surface will indicate the 

 pressure belonging to it, and the surface may be referred to as the surface 0, 1, 2 or 3. The 

 unit of pressure is that pressure which is produced by unit of force acting on unit of surface. 

 In order therefore to diminish the unit of pressure as in (5) we must diminish the unit of force 

 in the same proportion. 



(12) It is easy to see that these surfaces of equal pressure must be perpendicular to the 

 lines of fluid motion ; for if the fluid were to move in any other direction, there would be a 

 resistance to its motion which could not be balanced by any diff'erence of pressures. (We must 

 remember that the fluid here considered has no inertia or mass, and that its properties are those 

 only which are formally assigned to it, so that the resistances and pressures are the only things 

 to be considered.) There are therefore two sets of surfaces which by their intersection form 

 the system of unit tubes, and the system of surfaces of equal pressure cuts both the others at 

 right angles. Let h be the distance between two consecutive surfaces of equal pressure mea- 

 sured along a line of motion, then since the difference of pressures = 1, 



kvh = 1, 

 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 = \, 

 we find by the last equation 



« = 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 pi-essure p to a pressure {p—l) 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. 



