Mr maxwell, ON FARADAY'S LINES OF FORCE. 33 



(8) It is evident that the amount of fluid which passes any fixed surface is measured by 

 the number of unit tubes which cut it, and the direction in which the fluid passes is determined 

 by that of its motion in the tubes. If the surface be a closed one, then any tube whose ter- 

 minations lie on the same side of the surface must cross the surface as many times in the one 

 direction as in the other, and therefore must carry as much fluid out of the surface as it 

 carries in. A tube which begins within the surface and ends without it will carry out unity 

 of fluid ; and one which enters the surface and terminates within it will carry in the same 

 quantity. In order therefore to estimate the amount of fluid which flows out of the closed 

 surface, we must subtract the number of tubes which end within the surface from the number 

 of tubes which begin there. If the result is negative the fluid will on the whole flow inwards. 



If we call the beginning of a unit tube a unit source, and its termination a unit sink, then 

 the quantity of fluid produced within the surface is estimated by the number of unit sources 

 minus the number of unit sinks, and this must flow out of the surface on account of the 

 incompressibility of the fluid. 



In speaking of these unit tubes, sources and sinks, we must remember what was stated in 

 (5) as to the magnitude of the unit, and how by diminishing their size and increasing their 

 number we may distribute them according to any law however complicated. 



(9) If we know the direction and velocity of the fluid at any point in two different cases, 

 and if we conceive a third case in which the direction and velocity of the fluid at any point is 

 the resultant of the velocities in the two former cases at corresponding points, then the 

 amount of fluid which passes a given fixed surface in the third case will be the algebraic 

 sum of the quantities which pass the same surface in the two former cases. For the rate at 

 which the fluid crosses any surface is the resolved part of the velocity normal to the surface, 

 and the resolved part of the resultant is equal to the sum of the resolved parts of the com- 

 ponents. 



Hence the number of unit tubes which cross the surface outwards in the third case must 

 be the algebraical sum of the numbers which cross it in the two former cases, and the number 

 of sources within any closed surface will be the sum of the numbers in the two former cases. 

 Since the closed surface may be taken as small as we please, it is evident that the distribution 

 of sources and sinks in the third case arises from the simple superposition of the distributions 

 in the two former cases. 



II. Theory of the uniform motion of an imponderable incompressible fluid through a 



resisting medium. 



(10) The fluid is here supposed to have no inertia, and its motion is opposed by the 

 action of a force which we may conceive to be due to the resistance of a medium through 

 which the fluid is supposed to flow. This resistance depends on the nature of the medium, 

 and will in general depend on the direction in which the fluid moves, as well as on its velocity. 

 For the present we may restrict ourselves to the case of a uniform medium, whose resistance is 

 the same in all directions. The law which we assume is as follows. 



Vol. X. Paet I, 6 



