■A50 Mr. R. Haroreaves on 



addition to the terms there given a term ■ , 



vanishing for X infinite and r<l, but infinite for the upper 

 limit when r>l. The c-component of electric force, viz. 



2 N c: is infinite with N . For the external solution 



this infinite constant also appears, viz. : 



y 2 7V' C 2 -^(^-l) 7 , V / ^-\)(^-ir 



In (91) or 2(pL + 2jt/L') + 2<j> = 0, which is the condition 

 that V^ = should be satisfied, the variable terms disappear 

 correctly, but the constant in N remains and invalidates the 

 result. 



Consider now the character o£ the surfaces 



* 2 +y 2 , z 2 , 



a 2 + X + c 2 -X(r 2 -l) _1 ' 



where r>l. The polar axis diminishes from c to 0, while 

 the equatorial axis increases from a to <\/a 2 + c 2 (r 2 — 1). 

 As a consequence of this contrary movement each surface 

 from A, = to X = c 2 /(r 2 — l) intersects the conductor for which 

 A = 0, and there is mutual intersection of any pair of surfaces 

 outside the conductor. For the solution of a conductor 

 problem it is essential that there should be a series of equi- 

 potential surfaces not intersecting, and that one of these 

 should be the surface of the conductor itself. This condition 

 is broken, there are two potentials generally of finitely 

 different values at points on or outside the conductor, and 

 there is no field of constant potential within it. 



The envelope of the surfaces X consists of two equal cones 

 having a common base z=Q, % 2 -{-y 2 = a 2 + c 2 /(r 2 — l); the 

 vertices are on opposite sides of this base and the angle of 

 the cones is twice the angle arc sin 1/r or arc sin Y/io. The 

 cones also touch the spheroid (x 2 +y 2 )/a 2 + z 2 /c 2 = l, and the 

 contact lines separate the parts of the envelope belonging to 

 values of X greater and less than 0. These groups intersect 

 each other within the spheroid. 



Matters are not essentially different for the general case, 

 but one or two points in connexion with the geometry may 

 be of interest. In the isotropic case, as X falls from -{-co to 

 -co the order of types is ellipsoid, hyperboloid first of one 

 sheet then of two sheets, imaginary surface ; with the points 

 of division X = — c 2 , - b 2 , —a\a >b>c). In the motional case 



