ON FAEADAYS LINES OF FORCE. 163 



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 components. 



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 distri- 

 butions 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 



212 



