A respectively, and whose electromotive forces are e1 and e2, are connected in multiple arc, their positive terminals being joined at A (Fig. 104) and their negative terminals at B, the difference of potentials between the points A and B may be found from the following consideration. The two cells form a closed circuit, round which there will in general be a current which will traverse one of the cells, namely, that one which has the greater electromotive force, from the negative to the positive terminal (this may be called the natural direction), and the other from the positive to the negative terminal (or in the inverse direction). Let c denote the strength of this current; then, according to § 109, the difference of potentials between A and B will be less than the electromotive force, say e, of the stronger cell, and greater than that, e2, of the weaker cell, by the product of the current strength into the resistance of the corresponding cell. If E be this difference of potentials, we have accordingly E ei cri and E = e2 + C12. Or, eliminating c between these two values, B FIG. 104. 7° Now suppose the points A and B connected by a wire of resistancer (Fig. 105). This wire will be traversed by a current flowing from A to B of strength C = c1 + C2, if c1 and c2 denote the currents now flowing from B to A through the two cells respectively. As the result of this current, the difference of potentials between A and B will now be less than before: let it be denoted by F. Then, by the definition given at the end of $ 110, we have for the resistance of the single cell which would be equivalent to the given combination of two cells €2 B FIG. 105. To determine F, we may apply the same definition to the two cells taken separately; we thus get Remembering that C = c1 + 2, these equations give Putting these values into the expression for R, we have Hence it appears that when two conductors are connected in multiple arc, the resistance of the combination, or, in other words, the resistance of the single conductor by which the two given conductors can be replaced without altering the strength of the current in any other part of the circuit, is in all cases equal to the product divided by the sum of the separate resistances, whether these are simple metallic wires, or whether they are galvanic cells, or other arrangements which form the seats of electromotive forces. In the first case, the single equivalent conductor must not contain an electromotive force; in the second case, it must be the seat of an electromotive force E = (e1 r2 + е2 r1) | (r1 + r2). 112. Ohm's Law.—The nature of electric resistance, and the fact that the resistance of any given conductor has a definite numerical value characteristic of that conductor, were first clearly recognised by G. S. Ohm. The properties of the electric circuit were discussed by him from this point of view in a work published in 1826, and the relations that he established between the electromotive force or forces acting in a circuit, its resistance, and the strength of the current by which it is traversed, form the starting-point of all calculations involving these magnitudes. This relation is usually referred to as Ohm's Law, and may be expressed in symbols by the following equation E which may be applied either to a single conductor or to a complete circuit. R is the resistance of the conductor, or the total resistance of the circuit, as the case may be; E the electromotive force acting in the conductor, or the resultant electromotive force of the circuit ; and C the strength of the current. The unit, in terms of which resistances are commonly expressed, or the practical unit of resistance, is called an ohm, and is such that an electromotive force of one volt applied to a homogeneous metallic conductor having this resistance will maintain in it a current of one ampere. The ohm will be more fully discussed in the chapter on Electrical Units (Chap. xxx.), but we may state here that it is very nearly equal to the resistance at o° C. of 14.452 grammes of mercury in the form of a column of uniform cross-section and 106.3 cm. long. Ohm not only showed that the resistance of a given conductor in a given condition is a constant quantity depending on its material and dimensions, but he also showed that the resistance of a conductor of uniform cross-section s, and length /, is expressed by R = where p is a constant characteristic of the material, called its specific resistance. The full statement of Ohm's law must be taken as including this equation. 113. Specific Resistance.—If we put R1 for the resistance of a conductor of length 1 cm., and cross-section 1 cm.2, 7 and s in the above expression both become unity, and we have R1 = p, which gives us the definition of the specific resistance of a material as the resistance of a conductor of that material 1 cm. long and having a cross-section of 1 cm2. The shape of the cross-section makes no difference, so that, for the purposes of this definition, we may suppose that we take 1 cm. length along a square rod measuring t cm. across each way; such a piece would be a centimetre-cube, and its resistance would be the specific resistance of the material. But it is important to notice that, when specific resistance is defined as the resistance of a centimetre-cube, it is to be understood that the flow of electricity takes place between opposite faces and is uniformly distributed throughout the cross-section. The specific resistances of the metals increase with temperature, and therefore also the actual resistances of conducting wires. If L the resistance of a wire at o° C. is denoted by R, its resistance at to may be represented to a first approximation by R = R2 (1 + at), where a is a coefficient depending on the material. For several pure metals, the value of a has been found to be from 0.0036 to 0.0038, or nearly equal to the coefficient of expansion of gases. For alloys, the value of a is much smaller, and bodies are known for which its value is actually negative beyond a certain moderate temperature. Recent experiments by Professors Fleming and Dewar show that at very low temperatures, such as can be obtained by the evaporation of liquid oxygen, the specific resistances of the pure metals are very much less than they are at ordinary temperatures. The following numbers are taken from their results : The specific resistances of non-metallic liquids are enormously greater than those of metals, and they decrease with rise of temperature. A table giving numerical values for a number of substances, both metallic and non-metallic, is given at the end of this volume. CHAPTER XII. APPLICATIONS OF OHM'S LAW. 114. Current in a Simple Circuit.—Ohm's law enables us to calculate immediately the strength of the current that a given battery will produce through any conductor of known resistance which may be connected with its terminals. For instance, suppose the battery to consist of n cells connected in series, each having electromotive force e and resistance r', and that its terminals are connected by a wire of resistance r. Then the total electromotive force of the circuit is E ne, and the total resistance is R = nr' + r; consequently the current is = If the cells composing the battery, instead of being all identical, have various electromotive forces and resistances, e1, e2, and =r+ rí, rá, we shall have E e + e2 + and R ...9 rí+r2+ . . Again, if any of the cells are connected with their poles in the opposite direction to the rest, the values of e for them must be taken as negative—that is to say, E must be taken as the algebraic sum of the separate electromotive forces. The resistance of a cell comes into account in the same manner, whichever way it is connected in the circuit; hence the value of R is not affected by the reversal of any of the cells. 115. Geometrical Representation. The influence on the strength of the current of the electromotive force of the battery, and of the relative resistance of the battery and the rest of the circuit, can be conveniently expressed geometrically by the construction of Fig. 106. Here horizontal distances represent resistances, and vertical distances electromotive forces or differences of potential. MN represents the resistance of the battery, NO the external resistance, MK the electromotive force, and NL the difference of potentials between the battery terminals. The strength of the current is given by the slope of the line KO, |