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Full text of "Scientific Papers - Vi"

142 THE EFFECT OF JUNCTIONS ON THE between ra the radius of the inner and r' that of the outer conductor is, omission of eip{, z having the value proper to the section. On the negative side the c spending integral is (H^e-U" + K^*) log (r'/n), - rx being the radius of the inner conductor at that place. But whe conside'r the intermediate region, where electrostatical laws prevail recognize that these two integrals must be equal; and further that exponentials may be identified with unity. Accordingly, the first relabic (ff, + J5TO log (r'/n) = HI log (r'/r2). - - ..- ............... (1 In like manner the magnetic force in (14;), (16) is purely circumfere: And the circulations at the two sections are as JH,. — K^ and H2. But these circulations, representing electric currents which may be treatc steady,, are equal, we have as the second relation — (1 The two relations (17), (18) determine the wave propagated onward and that reflected Kl in terms of the incident wave H^. If ra = rl} we of course, H2 = Hl} JKl = 0. If we suppose r1( r2, r' all great and nearly equal and expand logarithms, we fall back on the solution for the two-dimensional already given. In the above the radius of the outer sheath is supposed uniform thro out. If in the neighbourhood of the origin the radius of the sheath cha from TI to ra', while (as before) that of the inner conductor changes from r2, we have instead of (17), while (18) remains undisturbed. In (19) the .logarithmic functions are proportional to the reciprocal the electric capacities of the system on the two sides, reckoned in each per unit of length. From the general theory given in the paper refe to we may infer that this substitution suffices to liberate us from restriction to symmetry round the axis hitherto imposed. The more ger functions which then replace logr on the two sides must be chosen ^ such coefficients as will make the circulations of magnetic force equal. generalization here indicated applies equally in the other problems of paper....(16)