however, vary enormously with the electrical conditions in the conducting gas, and for this reason a brief discussion of the phenomena of gas conduction is here given. ELECTRICAL CONDUCTION IN THE ARC, SPARK, AND DISCHARGE TUBE. dq Our first problem is to find a relation—a sort of generalized Ohm's law-between the current passing through a gas, the drop in potential and certain constants relating to the gas and the metal used as electrodes. This relation involves the solution of a differential equation containing an unknown function. This function I have constructed, but it gives a nonintegrable form to the differential equation. However, the particular case of steady current admits of a graphical solution (due to W. Kaufmanna) and covers a wide variety of the phenomena with which we are concerned. Consider a circuit containing an e. m. f. En, an ohmic resistance R, inductance L, capacity C, and a column of gas through which the drop of potential is expressible by the (experimentally determined) function eli) of the current. Equating energy supplied to energy taken up by the circuit, .di tieli) The final term, ie(i), represents the energy used up in the discharge in whatever manner the drop of potential e varies with the current. As an illustration, consider the energy used by a column of nitrogen 10 cm long and at 1 mm pressure, with platinum electrodes carrying 1 milliampere of current. The anode fall of potential is about 20 volts, the fall through the gas about 600 volts (60 volts per cm), and the cathode fall 260 volts. Hence the total energy used will be about 0.9 watt, of which about a third is used up at the cathode, two-thirds in the gas, and a negligible amount at the anode. If the column were shortened to 5 cm the energy used at the electrodes would be the same, while in the gas but half the previous amount of energy would be used up, and the total fall of potential would be 580 instead of 880 volts. In an ordinary illuminating arc carrying a current of 10 amperes, of the total drop in potential of approximately 50 volts, about half is at the electrodes, so that the luminescent gas receives about half the energy supplied. In the spark we may be reasonably certain that the energy losses are distributed in a manner intermediate between a Ann. d. Phys. 2, pp. 158–178; 1900. 2214-No. 3—0548 that of the arc and the discharge tube mentioned above. In every case it appears to require a greater expenditure of energy to force a (positive) current from a gas into a metal than from a metal into a gas. Now, the loss of energy at the electrodes is not within the electrode itself, nor is it in the adjacent gas outside; it is localized right at the surface separating the electrode from the conducting gas, and represents energy used in forcing the current from the metal to the gas or vice versa. Apparently all the electrode loss of electrical energy reappears as heat energy, about half being given to the gas and half to the electrode. In the spark with capacity, probably a large part of this energy is used up directly in vaporizing the surface layer of the metal, the discharge being so sudden that there is no time for the heat to be communicated to the adjacent metal or gas. In the arc a considerable part of this surface heat is conducted back and radiated from the solid electrode. The electrical energy equation above can only be solved when the function e() is known. This function may be constructed from the properties of its derivative in the following manner: since eli) has in general a single maximum and a single minimum, but no real roots, write for the derivative de=a (e-e) (e-e) di, where e, and e, are the ordinates of the maximum and minimum. By integration a function having the general properties of the characteristic curve for a gas, but which does not give a solvable form when substituted in the energy equation. When the current is steady the energy equation becomes Ev=iR+eli), from which the value of the current may be obtained (Kaufmann, I. c.) graphically, it being the abscissa of the point of intersection of the line E-iR with e(i). (The abscissa of the point of intersection of y=91.2) and y=92() is a root of the equation P(x)-92(x)=0.) Since in a conducting gas but little energy is stored in comparison with what is dissipated, the eli) in the last equation is the same as that in the general equation above. Some typical characteristic curves, e(i), are shown in fig. 1. The curve marked g shows the drop of potential as a function of the current in its general form. It is to be noted that: (1) the current starts with a brush discharge, during which the drop increases with increasing current; (2) there is a maximum drop at which the gas breaks down; (3) after the gas breaks down for a time the drop decreases as the current increases; (4) there is a minimum drop at which the drop is independent of the current, (5) followed by a region in which drop and current increase together. This form of curve may be realized in a gas at moderate pressures with the electrodes at a moderate distance apart. The curve a, fig. 1, is characteristic of an ordinary arc. The maximum is very high (say 20,000 volts), while the minimum is low, only about 50 volts. The downward sloping part of the curve is the portion usually observed, using but little series resistance and low e. m. f. To start an arc, like any other form of gas conductor, one must either (1) raise the e. m. f. above the maximum drop or (2) depress the maximum drop below the e. m. f.; the latter may be done by bringing the electrodes into contact, starting ionization by other means or lowering the pressure on the intervening gas. The curve b is characteristic of a brush discharge at high pressures between electrodes wide apart or pointed. The electromotive forces concerned are very large, while the currents are small. c is typical of very small distances and high pressures. It was obtained by Guthe and Trowbridge, using steel balls nearly in contact (107 cm apart) as a coherer. The maximum drop is only about one-fourth volt, while a Studied more in detail, this curve shows steps and other irregularities indicating saturation, ionization by impact, discharge not covering electrodes, etc., but which need not be considered here. Phys. Rev., 11, p. 29; 1900. the corresponding current is nearly half an ampere. All these curves are seen to be special forms of the type curve g, and all are capable of representation by the function e(0) above constructed. Given the characteristic curve for a gas under given conditions, then by means of the Kaufmann diagram (fig. 2) we may find what the current will be in a circuit containing an e. m. f. E, and an ohmic resistance R. The diagram is drawn for a circuit containing a gas at rather low pressure, with a moderately large e. m. f. controlled by considerable resistance. Keeping the e. m. f. E. constant, if the external resistance be increased, the line E-iR will take a greater and greater slope and intersect the eli) curve farther and farther to the left. At a certain slope it will intersect the curve in three places and the current may become intermittent. Finally it will become tangent to the under side of the gas curve and the heavy current will cease abruptly, leaving only the brush discharge corresponding to the upper intersection. Diminishing the external resistance, the current will not start again where it ended so abruptly, but only when the resistance line becomes tangent to the curve on its upper side. By taking a higher e. m. f., only single intersections will occur as the resistance is varied, hence the current will never become intermittent and the whole e(i) curve may be followed through experimentally, as it can not be when the e. m. f. used is just sufficient to break down the gas. Starka has worked out the relation between the critical minimum current for the arc (when the a Ann. d. Phys. 12, pp. 694–699; 1903. arc breaks) and e. m. f. and electrode metal. The relation to the density of the surrounding atmosphere may be seen from figs. 1 and 2, since as the pressure is diminished, curve a, fig. 1, is deformed into curve g. When a spark or Plücker tube is excited by an induction coil or transformer, it is the e. m. f. E, that is variable, while the resistance remains constant. In this case the E.-iR line moves parallel to itself, and the resultant current flowing through the gas has quite a different wave form from what it would have if the current were controlled by varying the resistance alone. Fig. 3, taken from the author's paper on Rectifying Effects, shows a typical curve for the current through a Fig. 3.-Alternating current curve through gases. gas upon which a sine alternating e. m. f. E (t) is impressed. Starting with the point a on the curve E (t) giving the e. m. f. as a function of time, the point c on the curve i(t) giving current as a function of time is obtained by means of the intermediate points E and b, assuming that the characteristic curve e() for the gas is of the form shown and that sufficient ohmic resistance has been used to give the line E-i Rthe slope indicated. Fig. 3 of course represents the current only in the ideal case of a circuit containing neither inductance nor capacity. When inductance (L) is added, instead of E-iR we must use the line di di a dt to find the current |