ON FARADAY'S LINES OF FORCE. 161 



motion no part of the fluid can flow across it, so that this imaginary surface 

 is as impermeable to the fluid as a real tube. 



(4) The quantity of fluid which in unit of time crosses any fixed section 

 of the tube is the same at whatever part of the tube the section be taken. 

 For the fluid is incompressible, and no part runs through the sides of the tube, 

 therefore the quantity which escapes from the second section is equal to that 

 which enters through the first. 



If the tube be such that unit of volume passes through any section in 

 unit of time it is called a unit tube of fluid motion. 



(5) In what follows, various units will be referred to, and a finite number 

 of lines or surfaces will be drawn, representing in terms of those units the 

 motion of the fluid. Now in order to define the motion in every part of the 

 fluid, an infinite number of lines would have to be drawn 'at indefinitely small 

 intervals ; but since the description of such a system of lines would involve 

 continual reference to the theory of limits, it has been thought better to suppose 

 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 quadri- 

 laterals 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 



VOL. i. 21 





