Page:On Faraday's Lines of Force.pdf/7

Rh 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 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