82 Mb maxwell, ON FARADAY'S LINES OF FORCE, 



the lines drawn at intervals depending on the assumed unit, and afterwards to assume the unit 

 as small as we please by taking a small submultiple of the standard unit. 



(6) To define the motion of the whole fluid by means of a system of unit tubes. 



Take any fixed surface which cuts all the lines of fluid motion, and draw upon it any 

 system of curves not intersecting one another. On the same surface draw a second system of 

 curves intersecting the first system, and so arranged that the quantity of fluid which crosses 

 the surface within each of the quadrilaterals formed by the intersection of the two systems 

 of curves shall be unity in unit of time. From every point in a curve of the first system let 

 a line of fluid motion be drawn. These lines will form a surface through which no fluid 

 passes. Similar impermeable surfaces may be drawn for all the curves of the first system. 

 The curves of the second system will give rise to a second system of impermeable surfaces, 

 which, by their intersection with the first system, will form quadrilateral tubes, which will be 

 tubes of fluid motion. Since each quadrilateral of the cutting surface transmits unity of fluid 

 in unity of time, every tube in the system will transmit unity of fluid through any of its 

 sections in unit of time. The motion of the fluid at every part of the space it occupies 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 diff'erent 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 continual 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 bounding 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 reservoir ; 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 emit- 

 ting 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 estimated 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. 



