430 Mr. R. Hargreaves on 



by (82) ; and this is the equivalent of the first of ($6). Again, 



3A(X) = 2{A(X)(A+;?XA a ) + 2AYX)(A'H-/XA ff )}, 

 or 



3A„ = 2{a(A +^XA a ) + 2a'(A' +pXA a )} ; 



which is the relation got by adding the second of (86) to 

 the first multiplied by XA a . 



Again, if we add the members of the triad (81) with 

 multipliers oty /3', we get by taking account of (80) the first 

 of the relations 



A a (* + X«) = A« 2 + B7' 2 + C/3' 2 + 2A'£V + 2B'«£' + 2C'«y' \ 



A«(a +X«') = A/3V + B«'/3 + Ca' 7 4-A'(/3 7 +a' 2 ) I (87) 



+ B'(*'ff + 77') + C'( 7 V +/9/3'). . .) 



The second is got in the case of 7' from the same triad by 

 the use of multipliers 7' /3 a'. 



§ 23. Combining now the first relation of (86) with (79) 

 we have 



2 |W# +^/j = 0, and therefore •?#= constant. . (88) 

 dX 2 / d\ d\j d\ ' 



To complete the solution for a conductor the constant is to 

 be found in terms of the total charge which is given by 

 «=f!2ZXdS 5 where (Imn) are direction-cosines of a normal. 

 Now 



7Ar deb ,dd> ,d4> d<j>( dX , ,d\ ,dX\ 



and if ot is a perpendicular from the centre on the tangent 

 plane, I = x-^— , ... ; thus 



2 J dX 0%V dx dij 2 dz / 



and quoting (78) 



•I 



d\ dX 



The constancy of r-~, r being the volume of u =1. is a 



J dX ^ " 



consequence of (85) and (88). The constant is fixed by 



taking X = 0, when J=l ; thus 



