162 ON FARADAY'S LINES OF FORCE. 



is determined by this system of unit tubes; for the direction of motion is that 

 of the tube through the point in question, and the velocity is the reciprocal 

 of the area of the section of the unit tube at that point. 



(7) We have now obtained a geometrical construction which completely 

 defines the motion of the fluid by dividing the space it occupies into a system 

 of unit tubes. We have next to shew how by means of these tubes we may 

 ascertain various points relating to the motion of the fluid. 



A unit tube may either return into itself, or may begin and end at differ- 

 ent points, and these may be either in the boundary of the space in which we 

 investigate the motion, or within that space. In the first case there is a con- 

 tinual circulation of fluid in the tube, in the second the fluid enters at one end 

 and flows out at the other. If the extremities of the tube are in the bound- 

 ing surface, the fluid may be supposed to be continually supplied from without 

 from an unknown source, and to flow out at the other into an unknown reser- 

 voir ; but if the origin of the tube or its termination be within the space under 

 consideration, then we must conceive the fluid to be supplied by a source within 

 that space, capable of creating and emitting unity of fluid in unity of time, and 

 to be afterwards swallowed ,up by a sink capable of receiving and destroying 

 the same amount continually. 



There is nothing self-contradictory in the conception of these sources where 

 the fluid is created, and sinks where it is annihilated. The properties of the 

 fluid are at our disposal, we have made it incompressible, and now we suppose 

 it produced from nothing at certain points and reduced to nothing at others. 

 The places of production will be called sources, and their numerical value will be 

 the number of units of fluid which they produce in unit of time. The places 

 of reduction will, for want of a better name, be called sinks, and will be esti- 

 mated by the number of units of fluid absorbed in unit of time. Both places 

 will sometimes be called sources, a source being understood to be a sink when 

 its sign is negative. 



(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 terminations 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 



