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 A (fig. 3) draw AP, making the angle O AP=a; produce OA to A', making A A' proportional to the added resistance r', 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 AP and A' P' will intersect. From the point of intersection Q draw QO 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 B 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 BB' (fig. 4) proportional to r, make the angles OBP and O B'P', on the same side of B B', equal respectively to B 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 A P 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 appara tus 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 4-7 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 M N 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 AP 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:

- 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 A C, E F 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 66 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. process is applicable to any number of conductors. Thus, let AB, BC, and CD (fig. 8) represent the separate resistances of three conductors connected in multiple arc. Making the con

The same

*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.

struction indicated above, BL represents the resistance equivalent to A B and B C in multiple arc. Draw L M' parallel to A B C D, make CM=BL, draw M M' at right angles to B C, join M' D, and draw CN perpendicular to CD. CN then represents the joint resistance of the three conductors.

Now consider a conducting-system such as that indicated in fig. 9, where two points, P and Q, are connected through three conductors whose resistances are respectively r, r1, and r1; and -let a battery of electromotive force e make part of the first conductor. The strength of the currents and the distribution of potential in the various parts of the system can be represented as follows:-Take OA (fig. 10) to represent r, from O draw OB and OC in opposite directions perpendicular to O A, make CC' equal to OC and perpendicular to it, draw BC' cutting A O produced in D; then O D represents the joint resistance of and . Draw A E perpendicular to A O to represent the electromotive force e, and join E D. Then tan ZADE represents the strength of the current in the battery and the sum of the currents in the two parallel branches. These may be obtained separately thus: let the point of intersection of E D and OB be denoted by F; then OF represents the difference of potential between the points P and Q or the electromotive force which is effective in the two conductors of resistance ri and In O A make O FO F and draw F B and F' C; we have then, for the strength of the currents in the conductors whose resistances are represented by OB and O C, tan ZOBF and tan ZO CF respectively.

Next let two of the conductors connected together at P and Q contain galvanic batteries, and, as before, let the resistance of the branch containing one battery be r, while that of the branch containing the other is r1, and let the electromotive forces be e and e, respectively. If the batteries are so connected that both tend to make the potential at P differ from that at Q in the same sense, we have an arrangement of which a special case is presented by Poggendorff's "compensation method" for the comparison of electromotive forces. To obtain a geometrical expression for the strengths of the currents in the various parts of the circuit in the general case (that is, without assuming that there is "compensation" in any branch), we may proceed in the following manner :-Take OA and O B in the same straight line (fig. 11) to represent the resistances r and r1 of the two branches including the batteries, and O C at right angles to A B to represent the resistance r2 of the third branch. In CO produced make OA, O A, and OB, O B, also draw A, A, equal and parallel to OA and B, B, equal and parallel to OB, and join A, C and B, C. Let A, C cut OA in N, and let B, C

=

cut O B in M. From A draw A E at right angles to A O to represent the electromotive force e, and draw E M cutting OC in E'; similarly, from B draw BF to represent the electromotive force e, and draw FN cutting CO (produced) in F'. Then F' E' FO+O E' represents the electromotive force which is effective in the conductor of resistance r, represented by O C, and A E-FE' and FB-F'E' represent the electromotive forces which are effective in the branches of resistance r and ri respectively.

III. Ordinates represent Strength of Current, and Abscissæ represent Resistances.

With this system of coordinates, electromotive force is expressed by the area of a rectangle. Thus, if a given battery produces a current whose strength is represented by the ordinate M m of the point M (fig. 12), in a circuit the resistance of which is represented by the abscissa O m of the same point, its electromotive force must be proportional to the area of the rectangle OM; and if the battery is "constant," the currents, represented by the ordinates M, m,, M, m2, and corresponding to the resistances denoted by the abscissæ Om, O mg, will be such that the areas of the rectangles O M, O M,, and O M, are all equal; and hence the characteristic property of the battery will be expressed by the curve which is the locus of the points M, M1, &c.-that is to say, by a rectangular hyperbola whose asymptotes are the axes of no current and no resistance. When the hyperbola characteristic of a given battery is drawn, it is of course easy by measuring coordinates to find what current would flow through a given resistance, or conversely to find what must be the resistance of the circuit in order that the current may have a given strength; but the difficulty of tracing an hyperbola with accuracy greatly lessens the practical utility of this method of calculation. Since, however, the asymptotes are fixed, each hyperbola is completely defined when one point of it is given; and, in like manner, when the corresponding values of current and resistance are known in any one case for a given battery, each is determined for any other case when the value of the other is given. Accordingly the actual drawing of an hyperbola is not necessary; for when one point is assigned, any other points corresponding to given problems can be easily found.

For instance, let the coordinates of the point M (fig. 13), referred to the axes OX and OY, represent respectively the

*BF must be drawn in the opposite direction to AE if, as supposed above, the batteries are so connected that the difference of potential between the points P and Q due to each battery separately is of the same sign