sci// coxir.Mi'ou.iRy .iitixcis- i\ I'livsiis \ i i.u 



timis from one sort <>f s«."lf-m.iiiu.iiniiit; (liscli.irno lo iinotlier, r.i;., 

 from rIow lo arc or from om- kind of ^;lo\v to anoilu-r when ci-rtain 

 critiral roiulilions are transi;rrssr(I (criliral coiulitiotis wliirh mav 

 tlK-msolM's ili'priul on the l>atti'r\' ami ihc rirniit as well as the con- 

 st.uUs of tin- spark-gap). There are disroiitiniiities of eurreiit and 

 diseontinuiiies of \oItai;e at these transitions, and al)riipl chanj;fs 

 in the \ isil>le appearance in the discharge; and at each transforma- 

 tion there is a rearrangement of the distribution of space-charge in 

 the gas. Hitherto we have encountered space-charge only in one or 

 two of its simplest manifestations, retarding tlte How of an electron- 

 stream across a vacuum, anil suddenly annulletl when positive ions 

 are mingled with the stream. .Now we ha\e to consider much subtler 

 and more complicated cases, in which the space-charge varies rapidly 

 in density and e\en in sign from one part of the gas to another, and 

 the field and potential distributions are utterly distorted by it; and 

 these distortions are essential to the life of the discharge. This 

 distribution of space-charge is indeed dominant; and so I will write 

 down some formulae which mav be used to describe it. 



.'). l)iiiRi:ssi()\ TO W'kiii. Down Somk Sp.\< e-Ch.\rge 



I-".gl ATIONS 



The fundamental cfiualion of the electrostatic field, known as 

 Poisson's equation, is 



^ -\ = -n-^ + -j-^ + -r^ = - 47rp (8) 



a.v- ay- dz- ^ ' 



in which l' represents the electrostatic potential, and p the xoluine- 

 density of electric charge. 



We consider only the mathematically simplest case in which all 

 variables are constant over each plane perpendicular to the .r-axis, 

 and so depend only on the coordinate .v; as for example near the 

 middle of an exceedingly wide tube with the .v-axis lying al(jng its 

 axis. In this case Pois.son's equation is 



dH' dX 



in which A' represents the potential-gradient, or field strength with 



sign reversed.'* The value of X is determined at all points when the 



" Field-gradient is therefore, proportional to space-charge with sign reversed, 

 and rice versa. Positive fieUI-graditnt implies negative space-charge; negative field- 



