equal to them at B and B'. If, further, the points B and B' be taken so that CB. CB'=CA.CA', the circle of radius CP will Fig. 8.

D

B

A

be a common line of flow in each of the systems due to C, A, and A' and to C, B, and B', and will therefore remain a line of flow in the resultant system (see fig. 9) due to A, A', B, and B'. Moreover, since these four points are placed so that one circle can be drawn through them all, it is easy to see that this circle will be another flow-line of the resultant system; for the four poles may be grouped in two ways into two pairs of equal opposite poles (either A, B and A', B', or A, B' and A', B), each of which would separately give this circle as a line of flow. If the straight lines AB and A'B' are drawn and produced to their intersection at D (fig. 8), a circle drawn about this point as centre with radius = √DB.DA= DB'. DA' will be an equipotential line common to the systems due respectively to the source and sink at A and B, and to those at A' and B'; consequently this circle is also an equipotential line of the resultant system due to all four poles. Another equipotential line of the resultant system would be the circle (in this case imaginary) drawn with the point of intersection of AB' and A'B as centre so as to cut each of these straight lines in points harmonically situated with respect to their extremities.

Again, if four equal poles be taken situated as in the last example, but so that poles of the same sign are diagonally opposite each other-in other words, if the signs of either A and B or of A' and B' be interchanged, the circle passing through the four poles will still be a flow-line of the resultant system, but C, as well as D, will now be the centre of an equipotential circle, while the imaginary circle will be a line of flow.

37. Resistance.-Equation (17) gives, for the difference of potential of any two points whose respective distances from the several poles are r1, 72, 73, ..., and r', r', ... the value

since, for each pole, there are the corresponding terms and Q, log. If there are altogether K poles, k bein sign and of the opposite sign, where k is not less and if, further, they are all of the same strength, the of electricity crossing each complete equipotential lin of time is kQ; consequently the resistance of the port sheet lying between the equipotential lines which pas the given points is

This formula is in principle quite general; but the application of it in actual cases requires that we sh the position of the poles from which the distances r1 to be measured; and these cannot be ascertained (or a by elementary methods), except for comparatively few cases; for although it is comparatively easy to det equipotential lines for a given set of poles, the invers of finding the distribution of poles required to produ tential lines coinciding with two given curves on the sheet, presents in general very great mathematical and has hitherto received only partial solutions.

38. We will give here, in the first place, the a application of the general formula to the case of two two equal sinks at the angles of a quadrilateral in circle and so placed that unlike poles are diagona each other. This is the arrangement shown in figs. 8 a A and A' may be taken as sources and B and B' as s evident that the equipotential lines very near the pol consist of two branches, one of them surrounding on sink), and the other surrounding the other source ( also that they will be very approximately circles havi at their centres. Hence, if the sources and sinks a four circular electrodes, whose common radius p is a tion of the distance between any two of them, and i placed that the distances of their centres A, A', B, a fixed point C fulfil the condition CA. CA'=CB is equivalent to saying that their centres are on the ci of one and the same circle), we may without serious the circles of contact between the electrodes and the sheet as forming together a pair of equipotential line

a distribution of poles as that referred to at the beginning of this paragraph. Then, taking points on the circumference of the circles round A and B as the points to which the values of ☛ and ' in equation (18) respectively refer, we have approximately (AB)2. AB'. A'B ; p2. AA'. BB'

Σ (log) = log

and since for the case supposed k=k=2, the resistance of the part of the sheet extending between the pair of circles round A and A' and the pair round B and B' is represented with similar approximation by

(AB). AB'. A'B

.log

(19)

If a circle be drawn with the centre C and radius

CPCA.CA'= √CB. CB',

this circle will coincide, as already pointed out (§ 35), with a flow-line of the system due to the combined action of the four poles. Consequently no electricity passes into or out of this circle; and therefore the whole of the electricity supplied by the source A flows to the sink B inside the circle (fig. 9), while the whole of what is supplied by the source A' flows to B' outside the circle. It follows, since the sources A and A' were assumed of the same strength, that the resistance offered by the part of the sheet lying within the circle of radius CP to the flow of electricity between the electrodes A and B is the same as the resistance of the part of the sheet lying outside this circle to the flow of electricity between the electrodes A' and B'. Hence also the resistance of a disk bounded by the circle in question and containing the two electrodes A and B is equal to twice the resistance of the entire sheet to all four poles, and is therefore represented by

1 R' = 2πκδ

(AB)2. AB'. A'B

(20)

which is the formula referred to in § 1 (page 387) as having been given by Kirchhoff for the resistance of a circular disk with two small circular electrodes anywhere upon it.

39. As already stated, this formula is only approximate, and in certain special cases it entirely fails. For instance, if one (or both) of the poles passes to the edge of the disk, then, in order that the circumference may still continue to be a line of flow, the second pole of the same sign must coincide with it; consequently in such a case AA' or BB', or both, will vanish, and the expression for the resistance fails by becoming infinite. The reason evidently is that equation (20) was got by assuming p to

be very small in comparison with any of the distances betwe the poles; and this can no longer be true when any two of th coincide. By slightly modifying the notation, we can obtain expression which does not fail in the same way. Thus, the points to which the values of r1, r,... refer be resp tively a point F where the straight line A B cuts the equipot tial circle round A, and a point G where the same line cuts equipotential circle round B. Then we have AF=BG=p, the expression for the resistance of the sheet to the current f all four poles becomes

while the resistance of the disk bounded by the circular f line may be written

In this form the expression for R' admits of extension to special cases. For example, let the pole A pass to the edge disk, while B goes to the centre; then A' will coincide w and B' will go to an infinite distance. We have then se AG=BF A'G= the radius of the disk (=P), also A'F= the resistance to the flow between the pole at the edge a pole at the centre of the disk becomes

If both poles go to the edge, A' coincides with A, and B' the system being reduced to two equal opposite poles at a AG=BF=2a from each other. In accordance with t resistance of the whole sheet becomes

ρ

which is the value already given for the same case tion (8).

40. If, all else remaining as before, the signs of the and B' are interchanged (that is, if A' becomes a sink source), the circle drawn with the centre C and radius ✔ becomes an equipotential line instead of a line of flow, t with centre D and radius DB. DA remains an equ line, and the only circular lines of flow are the circle th four poles and the imaginary circle (§ 36) whose cen intersection of AB' and A'B. The resistance of the sh flow of electricity between this new arrangement of pol by interchanging in (19a) the signs of the radii vector

The equipotential circles with the centres C and D being common to the system due to the source at A and sink at A', and also to that due to the sink at B and source at B', and having equal but opposite potentials when taken as belonging to either of these systems separately, will, in the system due to the four poles, form the two branches of the line of zero potential. The other equipotential lines of the system due to the four poles consist also each of them of two branches, both of which never lie within the same one of the two circles in question. From this it follows that each circle divides the sheet into two equiresisting portions; and consequently the resistance of each of them is AG.BF.A'F. B'G p2. A'G. B'F

R" =

1 Σπκδ

log

*.

(22)

41. Since half the lines of flow due to each pole lie within the circle drawn through them all, the resistance of the portion of the sheet bounded by this circle is (§ 25) twice the resistance of the unlimited sheet; consequently it is equal to R' or to R" according to the arrangement of the poles.

42. It was pointed out in § 39 that the resistance of the part of the sheet outside the circular flow-line with centre C, to the flow from the source A' to the sink B', is the same as the resistance of the part inside this circle to the flow from A to B. Accordingly the value of equation (20) remains unaltered when A' and B are put for A and B, and vice versa, and at the same A'B'2 time p2 (or PAP) is replaced by PAPPAPBAB

The circles round the two sources then coincide approximately with the two branches of a single equipotential line; and the same is true for those surrounding the sinks.

Similar remarks are applicable to equation (22).

Experimental verifications of some of the conclusions here arrived at will be given in Part II. of this communication.

*It may be noted that by adding together the values of R' and R" we BF 2

1

get 2-log (AGF), which, written in the simpler form

Σπκδ

is the resistance of a circular disk on whose edge the poles A and B are placed (see equation 13).

Errata in No. 326.

Page 395, line 3, for in read on.

398, line 17, for from line read from one line.