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where E stands for electromotive force, C for strength of current, and R for resistance, we have at once suggested three different modes of geometrical construction by means of rectangular coordinates, the coordinates representing in the three cases respectively (1) electromotive force and strength of current, (2) electromotive force and resistance, and (3) strength of current and resistance.

I. Ordinates represent Electromotive Forces, and Abscissæ

represent Strengths of Current.

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This system, according to which resistance comes to be represented by the slope of a straight line (that is, by the tangent of the angle which the line makes with the axis of abscissæ), was lately employed by M. Crova* for the discussion of experiments relating to the degree of constancy possessed by so-called stant" galvanic batteries; and its application to several other problems, including some of those treated by other methods in this paper, has been still more recently† pointed out by the same author. It is therefore not needful to discuss it further in this place.

II. Ordinates represent Electromotive Forces, and Abscissa

represent Resistances.

This system was used long ago by Ohm‡, and has been frequently employed since his time, though perhaps chiefly by practical electricians§. The following examples may serve to illustrate its application to questions connected with galvanic circuits in which there is a constant electromotive force.

Let OA (Plate VIII. fig. 1) represent the electromotive force of a battery, O B the resistance of the battery, and BC the resistance of the remainder of the circuit, this being made up of simple metallic conductors in which no additional electromotive force acts, then the slope of the straight line A C, or the tangent of the angle A C O, represents the strength of the current. It is obvious, by a glance at the figure, that the strongest current that the given battery could produce would be obtained by making the external resistance BC equal to nothing, and that it would be represented by the slope of the line A B, or by

* Comptes Rendus, 6th April 1874, vol. lxxviii. p. 965. † Journal de Physique (Sept. 1874), vol. iii. p. 278.

Die galvanische Kette mathematisch bearbeitet, 1827.

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See, for example, R. S. Culley, Handbook of Practical Telegraphy,' 1874, pp. 352-355; Latimer Clark, Elementary Treatise on Electrical Measurements,' 1868; G. K. Winter, "On Earth-Currents," Journal of the Society of Telegraph-Engineers, vol. ii. pp. 89–101; &c.

tan ZABO. Also it is evident that, if the external resistance is increased by equal amounts CC,, C, C,,... each equal to BC, the strength of the current, as denoted by the slope of the line drawn from A to the points C, C1, C2 . . ., diminishes by smaller and smaller amounts for each equal increment of resistance, and that it would not vanish for any finite value of the resistance.

If any electromotive force acts in the part of the circuit external to the battery, its effect on the strength of the current can be represented by drawing through C a line C C' parallel to O A, and of length proportional to the external electromotive force, upwards if this electromotive force is inverse, downwards if it is direct, and drawing the straight line A C' (fig. 2). If c be the point where this line cuts OC, tan 4 Ac O measures the strength of the current. Of course the effect of any electromotive force outside the battery could also be represented by a diagram such as fig. 1, if the line O A were there taken to represent, not the electromotive force of the battery, but the total resultant electromotive force of the whole circuit.

If a line be drawn from B (fig. 1) parallel to O A, the length BD, BD, BD2 . . . intercepted by the straight line through A, whose slope gives the strength of the current, represents the difference of potential between the terminals of the battery, or, in other words, the electromotive force which is effective in maintaining a current in the external conductor. The figure shows that this varies between a maximum (=0 A, the total electromotive force of the battery) when the external resistance is infinite (contact broken) and a minimum (=0), when the external resistance is nothing. If two values, BD and BD, of the externally effective electromotive force are known, which correspond respectively to two known values B C and B C1 of the external resistance, it is evident that the electromotive force and internal resistance of the battery will be given by drawing the straight lines C D and C, D,, producing them till they meet in A, and letting fall from A a perpendicular A O on CB produced: AO and OB then represent respectively the values required. Experimentally, the values to be given to B D and BD, could be found by direct measurement with an electrometer; or they could be got from the relation ecr', where e' is the externally effective electromotive force and c the current as measured by a galvanometer in a circuit of external resistance =r'.

From the above relations it is easy to deduce a construction, which may sometimes be of practical use, for finding the permanent resistance and electromotive force of a constant battery from two deflections of a galvanometer without using trignometri

cal tables. This construction requires to be slightly modified according to whether the instrument used is a tangent-galvanometer or a sine-galvanometer. It is as follows:

1. For a tangent-galvanometer. The battery is connected in simple circuit with a tangent-galvanometer, and the deflection a of the galvanometer is observed; then a known resistance is added to the circuit and the deflection is observed again. Let the second deflection be denoted by a'. The following construction then gives the electromotive force of the battery (=e) and the permanent resistance of the circuit (=).

From any point A in the straight line O À (fig. 3) draw A P, making the angle O AP=a; produce O A to A', making A A' proportional to the added resistance, and from A' draw A' P', making the angle OA' P'd and on the same side of OA as AP. Since a' is less than a, the straight lines A P and A' P' will intersect. From the point of intersection Q draw QỌ perpendicular to OA. Then OA represents the permanent resistance of the circuit, and QO represents the electromotive force, in terms of that electromotive force taken as unity which, if it acted in a circuit of unit resistance, would generate a current capable of causing a deflection of 45° on the particular galvanometer employed.

2. For a sine-galvanometer.-Let ẞ be the deflection observed when there is no extra resistance, and B' the deflection when an additional resistance has been inserted in the circuit. Make B B' (fig. 4) proportional to r', make the angles OBP and O B'P', on the same side of B B', equal respectively to ẞ and B', and let BP and B' P' intersect at the point Q. Draw QO to bisect the external angle BQ P', and cutting B' B produced in O; then O B represents the permanent resistance of the circuit, while the radius of a circle drawn with the point O as centre so as to touch the straight lines BP and B' P' measures the electromotive force, the unit of measurement for the latter being the electromotive force which, in a circuit of unit resistance, would give a current strong enough to deflect the galvanometer used through 90°*.

If the constructions indicated above are carried out for several different values of the external resistance, it is clear that, with a strictly constant battery, lines drawn according to the same rule * It may be worth while to point out that any galvanometer may be used as a sine-galvanometer, even though it is not provided with a graduation to show the angle through which it has been turned. It is only needful, after setting the instrument so that the zero of the scale is exactly below the needle while the current is passing, to interrupt the current; the needle then swings away from the zero-mark, returning to the magnetic meridian; and the angle now indicated by it is the angle through which the galvanometer has been turned from the meridian.

as AP and A' P', in the case of a tangent-galvanometer, will all pass through the same point, and that, in the case of a sinegalvanometer, all lines drawn in the same way as BP and B' P' will be tangents to the same circle.

· These constructions are so simple and can be so quickly made with sufficient accuracy, that by means of them the effect of altering the resistance of a circuit or the number or arrangement of the cells of a battery can be exhibited to a class by the help of actual measurements made during a lecture; but in order to make the process still more rapid, I have had an apparatus made, which may be called "A Galvanometric Slide-Rule," whereby, when two deflections of a galvanometer have been observed corresponding to a known difference of resistance, the permanent resistance and electromotive force of a battery can be ascertained in the course of a few seconds. The general arrangement of this apparatus is shown in fig. 5, which is drawn to a scale of about. MN is a wooden base about 155 centims. long by 15.2 centims. wide, and 47 thick, with a groove in the upper surface, of the shape shown in fig. 5a, running from end to end. On the vertical side shown in the figure there is a scale 150 centims. long, divided into millimetres and numbered towards right and left from zero at a point A 50 centims. from M and 100 centims. from N. Exactly above the zero mark of the scale is a small brass stud, the axis of which passes through the centre of a small graduated quadrant of 12 centims. radius. Another, similar quadrant, A', with a brass stud at the centre is attached to a small board which slides on the upper side of MN and can be clamped by a screw in any position between A and N. There is also a vertical scale 65 centims. long, divided into millimetres and numbered from the bottom upwards, which is fastened to a sliding piece C, whereby it can be clamped in any position between M and A. The faces of the vertical scale and of the quadrants A and A' are flush with the face of the horizontal scale M N. Two thin silk cords, stretched by small weights, are passed over pulleys at P and P', and attached one to each of the pins at the centres of the quadrants A and A'. The pulleys are so placed that the cords are very nearly in the same vertical plane as the scales M N and CQ. The apparatus is used as follows:-The pulley P is raised or lowered until the cord fixed at A shows, upon the corresponding quadrant, the deflection obtained on the tangent-galvanometer when no extra resistance is added to the circuit; the sliding piece carrying the quadrant A' is then moved towards right or left so that the number of centimetres in the distance A A' may be the same as the number of units of resistance (or, if more convenient, so that it may be a simple multiple or submultiple of this number)

added to the circuit in order to get a second reading of the galvanometer; then the pulley P' is adjusted so that the cord passing over it may indicate upon the quadrant B the galvanometer-deflection obtained after introducing the resistance corresponding to A A'; and lastly the vertical scale is moved so that a vertical line drawn through the centre of the divisionmarks and continued downwards to meet the horizontal scale at O may be exactly behind the point Q, where the cords A P and A'P' cross each other. We have then only to read off the horizontal and vertical distances, A O and O Q, to get the permanent resistance and electromotive force of the circuit*. To adapt the apparatus to use with a sine-galvanometer, it would be only necessary to replace the vertical scale OQ by a quadrant marked with concentric circular arcs, each differing from the next by 1 millim. in radius.

The following additional examples may be given of the same mode of treatment in connexion with allied problems :

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Development of Heat in a Galvanic Circuit.-Draw OC (fig. 6) horizontally to represent the resistance of the circuit, and OA vertically to represent the electromotive force; from A draw AC and also A D at right angles to AC; then OD represents the heat produced in the circuit per unit of time. It is of course to be understood that OA denotes not necessarily the electromotive force of the battery, but the resultant electromotive force of the whole circuit-that is, the algebraic sum of all the electromotive forces which act anywhere in it. If OB denote the resistance of the battery and BC the remaining resistance of the circuit, and if B A' be drawn vertically, A'E F horizontally, and A'D' at right angles to AC, EF will denote the heat generated in the battery, and B D' the heat generated in the external part of the circuit.

Strength of Currents in the different branches of a divided Circuit. -In considering this problem it is needful first of all to have a mode of representing geometrically the combined resistance of two or more conductors connected in " multiple arc." Let A B and BC in the same straight line (fig. 7) represent the resistances of two conductors taken separately. Draw A B' equal to A B and at right angles to it; join B' C, and through B draw a line parallel to A B' and cutting B'C in L; then B L represents the combined resistance of the two conductors. The same process is applicable to any number of conductors. Thus, let A B, BC, and CD (fig. 8) represent the separate resistances of three conductors connected in multiple arc. Making the con

*The quadrant A is made so that it can be turned aside to allow of the vertical scale being brought close up to the zero-point of the horizontal scale when low battery-resistances have to be measured.

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