OHM S LAW.] ELECTRICITY 43 matter how many meshes it may include, or what con ductors may branch off at different parts, we have where E is the u hole internal electromotive force, and 11 lf R 2 .... C lr C 2 ... are the resistances and current strengths in the different parts of the circuit. The first of these principles is simply the law uf con tinuity, and the second is got at once by applying equation (10). We give here an investigation of the currents ami potentials in a network of conductors. The method and notation are taken from Maxwell, vol. i. 280. Let A v A s , ... A,, be n points, con nected by a network of %n(n - 1) conductors (that being the number of different pairs of conductors that can be selected from the n). Let Cp,, Ep,, K M denote the current strength, internal electromotive force, and conductivity, i.e., the reciprocal of the resistance, for the conductor A p A,. at A p be P P , and the system there be symbols that " ctor A p A,. Let, moreover, the potential current Oi electricity which enters the It is obvious from our definitions of the ~ and, by the condition of continuity, that Qi + Q*+ +Qn 0. At the point Ap we have Now Heuce (a) becomes K p i( P! - Pp) + Kp^Pj, - Pp) + ....+ K pn (P n - Pp) = Kp!Ep!+ . . . . + Kp n Ep n -Qp . . . ( T ). The symbol K pp does not occur in this equation, and has no mean ing as yet. Let us define it to mean - (K fl + K P 2 .... Kjm), where K p p does not occur. Then we have Kp! + Kp 2 + . . . -f K + . . . + lpn = 0, .... (5) and, multiplying by P r - P r , Kpj(Pp - P r ) + . . . + K ;>p (Pp - P r ) . . . Kp,, (Pp - P r ) = . Adding this last equation to (7) we get In this equation the term whose coefficient is Kp r of course vanishes. By giving p all possible values except r, we get a set of tt - 1 equations to determine the n - 1 quantities Pj - P r , P 2 - P r , tc. Ilem-e if M rr denote the minor of K r r in the determinant K,,,,) ; 1 and if M rr p denote the minor of Kp, in Mrr, we have + &c (,, where of course E n and E 22 are zero, and il rrrp docs not occur. This expression is linear in the letters E and Q, and the principle of superposition holds, as we saw it ought to do in all applications of Ohm s law. Consider the particular case in which all the Qs and Es vanish, except E; m and E m i ( = - Ej m ), we then have the case of a linear circuit in which au electromotive force EJ M is introduced into A, A,. We get from (0 - P r - ~~ and Hence and = "M^ < M ^ r>a KpJv^Hj. Similarly, if (_ &, be the current in A<A due to au electro motive force Ep, in A M , we get . . . . (0) . 1 This determinant has many properties of interest to the mathe matical studcut; e.g., in oar notation iL, = M,. = M- Hn. v - M^,-r M* - y.. lr ic. ic. Now, since A is a symmetrical determinant, M rr j p = M rr pj , &c., and the expressions within brackets in (?;) and (9) are identical. Hence follows the important proposition : If an electromotive force equal to unity, acting in any conductor A, A., of a linear system, cause a current C to flow in the conductor A p A,, then an electromotive force equal to unity, acting in A P A, , will cause an equal current C to flow in A,A . If we suppose all the conductors of the system except AjA and A P A, removed, and A/A P and AA, joined by two wires, in such a way that for electromotive force unity in A/A W the current in A P A, is C then the conductivity of the circuit which we have thus constructed would be this might be called the reduced conductivity of the system with respect to A P A, and AjA m . When the expression within brackets vanishes, the conductors ApA, and A/A. arc said to be conjugate. Couju- The reduced resistance in this case is infinite, and no electromotive gate, con- force in AiA m , however great, will produce any current in ApA,, due-tors, and reciprocally. Similarly, we may prove that if unit current enter a linear system at At and leave it at A m , the difference of potential thereby caused between Ap and A, is the same as that caused between A, and A m , when unit current enters at Ap and leaves at A,. (See Maxwell.) The case of several wires forming a multiple arc very Multiplo often occurs in practice. Let AU, CD (fig. 20) be two parts of a circuit whose resistances are R and S, and let the cir cuit branch out between B and C into three branches of resistances We have YB - V c = KA = E 2 C 2 = R 3 C 3 , and 1 , R 2 , E 8 C, - Also V A - V D = V A - Y B + V B -Yc (R + p-fS)C, Hence current in each branch is inversely propor tional to the resistance, that is directly proportional to the conductivity ; and the reduced conductivity of the multiple arc is equal to the sum of the conductivities of its branches. These statements are obviously true for any number of branches. Some of the most important applications of the theory of linear circuits occur in the methods for comparing resistances. The earliest method for doing this consisted simply in putting the two conductors, whose resistance it was required to compare, into a circuit which remained otherwise invariable ; if the current, as measured by a galvanometer, was the same, whichever conductor was in the gap, it was concluded that their resistances were equal. The difficulty in this method is that the electromotive force and internal resistance of the battery are supposed to re. nain constant, a condition which it is excessively hard to fulfil. This difficulty can be avoided by using a differential galvanometer, or the arrangement of conductors called Wheatstone s bridge. The differential galvanometer differs from an ordinary one simply in having two wires wound side by side instead of a single wire. If we pass equal currents in opposite directions through the two wires, the action on the needle is zero, provided the instrument be perfectly constructed. If the currents are unequal, the indication will be proportional to the difference of the current strength. If the coils are not perfectly .symmetrical, but such that Resist ance measure incut. Di tier en- tial gal vano
meter.