QR, or RR, and since these quantities are equal to each other, it follows that the lines P'Q'R', PQR, and P, Q, R, are consecutive flow-lines of a system which divides the conducting sheet into portions each of which conveys an equal current.

It is important to observe that the reasoning employed here is general, and not limited to the special case to which it has been applied. The general conclusion to which it leads may be thus stated:-If similar * systems of lines of flow be drawn, corresponding to each of two separate systems of sources and sinks, the lines of flow which would result from the combined action of the sources and sinks of both systems will be obtained by drawing curves through the alternate angles of the quadrilaterals produced by the intersections of the two primary systems of flow-lines, in directions concurrent with both the primary flowlines that intersect each other at each angle.

Let

11. The method which, as we have seen, allows the flow-lines for two equal opposite poles to be drawn, also enables us to deduce very easily their general form. Let a be the constant angle between consecutive flow-lines of the pencil diverging from A and of that converging to B. Then, evidently,

ZAPB ZAQ B=LAR B= ... =na,

where n is a constant integer. Also

ZAP,B=ZAQ, B= ... = (n−1)a.

Hence the lines of flow due to one source and one sink of equal strength are arcs of circles passing through the poles, each one differing from the next by a constant change (a) in the angle which the radii vectores from the poles make with each other; or, what comes to the same thing, they are arcs of circles cutting each other at the poles with a constant difference of angle equal to the constant difference of angle (a) between the rectilinear flow-lines which either the source or the sink would produce by itself.

12. The whole number of flow-lines is therefore

2π

α

or the

same as the number of lines leaving the source or entering the sink when either of them is by itself in the sheet. This is evident also if we consider that infinitely near to either pole the effect of the other vanishes in comparison, and therefore the

* By similar systems is here to be understood systems such that the total flow between any pair of consecutive lines of the one set is the same as that between any two consecutive lines of the other set.

lines close to each pole are the same in all respects as they would be if the other pole were absent.

13. The circular form of the flow-lines for the case we are considering can be demonstrated in various other ways. We will give here one additional proof, on account of its great simplicity.

The flow through any point P due to a source at A and a sink at B, being the resultant of the currents through the same point due to A and B taken separately, will be represented in strength and direction by the third side of a triangle whose other two sides represent the currents from A and towards B, respectively, in strength and direction. But the strengths of the two component currents are inversely as the distances AP and BP respectively (§4) hence the following construction (fig. 2). From B draw BT parallel to AP, and make its length a third proportional to AP and PB; then PT gives the direction of the flow at P, and its length is proportional to the strength of the current at P. The similarity of the triangles APB and PBT gives the angle BPT equal to the angle PA B; and consequently (converse of Euclid, III. 32) the locus of P is a circle through A and B.

:

14. It was shown in § 10 how a system of lines of flow can be traced out by successive points. To be able to draw them continuously with compasses we only require to know the posi tion of the centres; and these are easily found from the following considerations. Since the circles of which the flow-lines are arcs pass through the poles A and B, their centres lie in the straight line at right angles to A B, through O its middle point. If C (fig. 3) be the centre of the circle which gives the flow-line through any point P, the angle at C is equal to the angle at P -the angle characteristic of the given flow-line; and therefore the angle O AC is the complement of the angle at P. Putting AB=2a, we have

Let the number of lines to be drawn be m, then the constant difference between the angles contained under consecutive lines will be

and the several lines will be given by making the angle at the circumference successively equal to

where the values О, π, -π, and 2 are represented by the straight line through A and B, and negative angles indicate flowlines passing in the lower side of AB. But, as is evident from the figure, the same circle gives the two flow-lines whose characteristic angles are na and —(π-na); hence the number of circles to be drawn or of centres to be found is equal to half the number of flow-lines; and we need only consider those characterized by angles between 0 and π, of which the complements are

Consequently the required centres are obtained by drawing lines from A making the above angles with A B, and letting them intersect the perpendicular to A B through O; or, without measuring angles, by laying off from O in both directions along the perpendicular to A B lengths proportional to

Plate IX. shows a system of lines of flow for which a=20°. 15. The strength of the current at any point P due to a source at A and an equal sink at B is represented in the construction given in § 13 by the length of the line PT (fig. 2); but by the similarity of the triangles PBT and A P B,

Hence, for given poles at a given distance apart, the strength of the current is the same at all points of the sheet for which the product of the distances from the two poles is constant; or the loci of equal flow are a system of lemniscates*.

* The strength of the current at various parts of a circular disk with equal opposite poles on the edge was examined experimentally by Kirchhoff, by suspending a small magnet, with a mirror attached, close above the disk, and was found to agree closely with the results of calculation. Mach has since given a more direct proof that the lines passing through points of equal flow are lemniscates. A disk of silver-leaf was coated with a thin film of wax; and on allowing a strong current to traverse it for a few moments, the wax was melted within a space bounded by a lemniscate and the edge of the disk (Carl's Repertorium f. experim. Physik, vol. vi. p. 11, 1870).

16. Equipotential Lines.-The flow-lines for the case of one source and one equal sink having been determined, the form of the equipotential lines is at once given by the consideration that the two sets of lines cut each other orthogonally (§ 2); and it is a well-known geometrical result that the system of lines orthogonal to the system of circles which, as has been seen, represents the flow-lines for this case, is another system of circles having their centres on the line through the poles, each circle cutting this line once internally and once externally in points situated harmonically with respect to its extremities. The simplest general expression for such a system of circles is the equation

7=c,

where c is a quantity which remains constant for each circle but varies from each circle to the next, and r and rare the distances from any point of the curve to the source and sink respectively.

These general properties of the equipotential lines are easily established by referring to the construction employed in § 13. We there saw that PT (fig. 2) is a tangent to the line of flow through P; and consequently it is normal to the equipotential line through the same point. If we produce PT to cut A B produced in C, we have the triangles B C P and P C A similar, and

CP2-CA.CB;

whence it appears that if tangents to the lines of flow be drawn from any point C in A B produced, the distance from C to the points where these tangents touch the lines of flow is constant and depends only on the distances of the point C from A and B respectively. Therefore, if a circle be drawn with centre C and radius C P, it cuts all the lines of flow at right angles, and is consequently an equipotential line. If it is only required to find the centre of the equipotential circle through any point P, the simplest method is to make an angle BPC equal to the angle BAP; then the point where PC and A B intersect is the centre to be found.

The similarity of the triangles B C P and PCA also gives

AP r AC

PB 7 CP

or the ratio of the radii vectores from the two poles is constant for a given circle, as already stated.

17. The above method suffices for drawing the equipotential lines through any number of given points, but not for placing them systematically (or so that the difference of potential between consecutive lines may be constant). For this purpose we may

have recourse to a process of superposition of the same kind as that employed (§ 10) for placing the lines of flow.

We have seen (§5) that the equipotential lines for a single pole are concentric circles, and that the radii of consecutive circles form a geometrical progression. To find the system of equipotential lines for two equal opposite poles, it is only needful to draw for each pole separately a system of equipotential lines with the same difference of potential between any one line and the next as it is intended that there should be between the lines of the combined system, and then to draw lines through the intersections of the two overlapping systems of circles thus obtained, taking care, in going from one intersection to the next, that the changes of potential are in opposite directions for the two primary systems taken separately. Thus, let the lines 1, 2, 3, 4 (fig. 4) represent portions of equipotential lines due to a source at A; and 1', 2', 3', 4' portions of equipotential lines due to a sink at B; and let the potential of the line 1 be V, and let that of the line 1' be V'; further, let the change of potential in passing from any one line to the next in either system be v, so that the potentials of the lines 2, 3, and 4 are V-v, V-2v, and V-3v respectively, and the potentials of 2', 3', and 4' are V'+v, V'+2v, and V+3v respectively. Then, at the points where 1 and 1, 2 and 2', 3 and 3', 4 and 4' respectively intersect each other, the potentials will be the sums of the potentials of the intersecting lines; and therefore the potential at all these points is the same, namely V+V'. Consequently P, Q, R, and S are points on the same equipotential line. Similarly it follows that P, Q, and R, are points on the line whose potential is V+V+v; and Q', R', and S' points on the line whose potential is V + V'-v. We thus get the potential of the resultant equipotential lines differing by the constant amount v, which is the same as the difference of potential of the lines of the two primary systems.

It is evident from this that any two systems of equipotential lines whatever, which have the same constant difference of potential, can be compounded, so as to give a single resultant system, by tracing lines through alternate angles of the quadrilaterals produced by the mutual intersection of the lines of the two systems, and also that the constant difference of potential between the lines of the resultant system will be the same as that between the lines of each of the component systems.

18. Let μ be the common ratio of the radii of the equipotential circles of the two primary systems considered in § 17, and let the radius of the circle 1 be " and that of the circle l' be μm (see § 8); the radii of the successive circles of the one set are then μ+1, μ+2, ..., and of those of the other μm+1, μm+2,

Phil. Mag. S. 4. Vol. 49. No. 326. May 1875. 2 E