an Ellipsoid at an External Point, 461 



We cannot from these six equations express Aj, A2, Bj, B2, 

 Ci, C2 in terms of the other four coefficients, because on 

 comparing the sums of the two sets of equations we get 

 E = 30( A + B + C) ; therefore we must seek some simple rela- 

 tion amongst the coefficients that may be used as another 

 equation. The coefficients are 



A=V + l2«^ + l3a2 + I,, 



Ai = 1 5Iia2/^ + UM^'' + /3^) + Uc^ + W') + 3I4, 

 A2 = 15Ii«^;Q^ + l2(6a2/3^ + a^) + 13(2^2 +^2) + 31^^ 



from which the others may be written down. The simplest 

 relation that suggests itself arises from adding the three differ- 

 ences Ai ~ A2 + Bi — B2 + Ci — C2 ; this is equal to 



Also A = P/: B^Pe^^ C = P6% the accents having the 

 meaning already explained. Thus we get 



" f Ai= P„+ P/-3F/'- Pe'", 



fA,= -P„-3P/+ P/'- Pe'", 



§Bi= P„- P/+ P/'-3Pe'", 



§B,= -P„- Pe'-3P/'+ Pe'", 



fCi= Po-3Pe'- Pe"+ Pe'", 



fC3=-Po+ Pe'- Pe"-3P/". 



These have to be substituted in Qg, with the following values 

 for the integrals : — 



J^%n= ^-^-^ Ma^ : ^xYdin= ^^^Ma%^ : 



M 



The result is 



+ P/^' e\ el{e\ - ^|) + Ye\ e\ e'l } . (6) 



The sequence of the terms (2), (4), (6), which complete the 

 series (1) as far as written down, is very remarkable, and sug- 

 gests the idea that possibly an expression might be obtained 

 for the general term. When the ellipsoid is one of revolution, 

 so that c = 6, and e^=-G^ — a^, 



r \ 3 . 5 ■ r^ "^ 5 . 7 ■ »•* 7 . 9 ■ r" "^ • ■ • / • 



