346 REPORT — 1872. 



consequently we shall have x 



2.{K,A'(G,,,J,.,,-J,,,G,,)} = 0. 

 This equation must hold good for all values of c ; wherefore, putting 



we shall have 



Ki,"Pi+K2.„p,+K3,„p3+ .... +K,,„jj„=0. 

 These equations ^iyq p^=p^= .... ==p„=0, 



«^ 2,G^,A,c=2,G^,.J...<.. (1) 



Now if we put 



then we shall have 



J.,C=V.,Ar„ (2) 



JV,c = Ke+2o'e,,,.K^.,, (3) 



The first of these formulae may be proved thus : 



«"^'^^ 2,G,,/.',c=S,G^',A«> 



therefore ' S,S,G,, ,K _ , J^, ,= S,S,G , /,. ,K ^ , ; 



that is (see equations B), J^, ^, = 2,2,G^, /^^ ^ . K^^ ^„ 



Also, for the second of these formiilaj, since by equations (2), section 6, wo 

 have 



^.-(■^i-, 0^1/, c' ~ ^v, c-^'i/, cO = ^> when c and c' are unequal, 



IT 



= K, when c and c are equal ; 



, hence 2,{ 2,(G,, ,K,_ J j;_ ^„ - 2,( J,, ,G,, JK',, ,,} 



_ ^p 

 ~ 2 "'. "'• 



