Nature . be the energy at the time t, a the rate at which itis absorbing the work done in the discharge tube; theenergy may be supplied to it from the Rontgen radiationin the tube or from the corpuscles which come intocollision with the atom, a will be proportional to therate at which the electric field producing the discharge isdoing work in the neighbourhood of the atom we areconsidering; it will thus be proportional to the productof the electric force and the flux of corpuscles in thisneighbourhood. Let us suppose that the system radiatesenergy at a rate proportional to E, say equal to /3E,
Nature . be the energy at the time t, a the rate at which itis absorbing the work done in the discharge tube; theenergy may be supplied to it from the Rontgen radiationin the tube or from the corpuscles which come intocollision with the atom, a will be proportional to therate at which the electric field producing the discharge isdoing work in the neighbourhood of the atom we areconsidering; it will thus be proportional to the productof the electric force and the flux of corpuscles in thisneighbourhood. Let us suppose that the system radiatesenergy at a rate proportional to E, say equal to /3E, thenwe have dE/dt = a-$E,or E = a//3(i-e-0Oif E = o when ( = o. Consider two different systems, A and B, in the sameatom; let E,, a,, ,8, ; E, a2, #2 be the values of E, a, 8for the systems A and B respectively. Et = aJPl (i-e-/V), E2 = aJP2(i-e-fl2/).Now suppose that the system A is one that does notabsorb much, but also does not radiate much, while Babsorbs a great deal more than A, but radiates still. more in proportion, so that o,>a, but aJ0v>a,!82, thenultimately E, is greater than E„, but at first E2 is greaterthan E,. The curves A and E, Fig. 4, represent thevariations of E1 and E, with the time. Suppose, now, that systems A and B become luminouswhen the internal energy is equal to W. It is not neces-sary to assume that the critical amount of energy is thesame for the two systems; the assumption is only madeto simplify the diagram ; it will be seen that the argumentwill apply if the critical amounts of energy are differentin the two cases. Now consider, first, the case when the rate at whichwork is being done in the tube is so small that thoughaJ/3, is greater than W, ct„/5, is less than W, the caserepresented in Fig. 4 ; here system A acquires the amountof energy necessary to make it luminous, while systemB does not; thus in this case the spectrum of the gaswould show the lines corresponding to A, but not thoseof B. Suppose, now, we increase the rate at which
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