392 Royal Society: — 



this as a tension r, the components at one end of ds will be 



—t(Iv : ds, —rdy : ds, — rdz : ds, 

 and those at the other 



(rdv :ds) + d(Tdx:ds) . . . , 

 the algebraical sums of which are 



d(jccd : ds), d(rdy : ds), d(rdz : ds) ; 

 and the equations of equilibrium then become 



C(Ydz -Zdy) + d( T d,v:ds)=0, (1) 



C(Zdx -Xdz)+d(rdy:ds)=0, (2) 



C(Xdy - Ydcc) + d(rdz :ds)=0; (3) 



taking s as the independent variable and multiplying by dec : ds, dy : ds, 

 dz : ds respectively, and adding, we obtain dr=0, or r = constant. 

 A gain, multiplying by X, T, Z and adding we obtain 



Xd\v:ds 2 + Yd 2 y:ds 2 + Zd 2 z:ds 2 = 0, .... (4) 

 which expresses that the absolute normal (or normal in the oscu- 

 lating plane) is perpendicular to the resultant magnetic force. 



In the case of a uniform tint, X, T, Z will be constant. In- 

 tegrating (4) and putting i for the angle between the tangent 

 and the lines of magnetic force, we find 



Xdx + Ydy + Zdz= Hds cos i, 

 so that the tangent line is inclined at a constant angle to the line 

 joining the poles. 



Again, the following combinations, (2) dz — (3)dy=0, (3) dx— (1) 

 cfe=0, (l)dy— (2)dx=0 give 



Cdx(Xdx + . .)-CXds 2 + T ($^ - C ^~)ds 2 ==0,&c., 

 Vis ds 2 dsds 2 ' 



or 



C(Ecos idx-Xds) + r(^^y - ^^f\ds=0, &c. 



Vis ds 2 ds ds ' 



Transposing, squaring, and adding, and putting p for the radius of 

 curvature, we obtain 



C 2 E 2 sin 2 i — r 2 : p 2 , or p= r : CE sin i, 

 which is constant. The curve is therefore a helix. Also the 

 radius of curvature of the projection of the curve on a plane per- 

 pendicular to the axis will be p sin 2 i, viz.=r sin^ : CE. 



"The value of r depends doubtless on the nature and pressure 

 of the gas, and perhaps also on the current ; but it must be the 

 same for equal values of C of opposite signs. Hence the handed- 

 ness of the helix will be reversed by reversing either the current 

 or the magnetic polarity. If the left-hand magnetic pole be north 

 (i. e. austral, or north-pointing), and the left-hand terminal positive, 

 the helix will be right-handed." 



The general nature of the phenomenon may therefore now be 

 described as follows : — " First we have the bright spark of no 

 sensible duration which strikes nearly in a straight line between 

 the terminals. This opens a path" for a continuous discharge, which 



