lloBKKTS — Tlic S'l/Dibolicdl Expression of EUvtiiKinls. 



71 



We have then, in general, on expanding the eliniinant in terms of Am, 

 (11) E{A„„ B„) = A„;'B,'" + A„r'X, + A„r'X, + A„r'X, + . . . 



+ A^-X„.^ + A,„X„., + {- l)"B„E{A,„-u B„), 

 where Xi, X2, . . . &c., are functious of the coefficients of both quaulics and 

 independent of A„,. 



We now introduce two new operators, which we denote by Q and w, and 

 define as follows: — 



If m be equal to or greater than n, we write, where r = m - n, 



(12) 



^'- = -^0 TIT + Bi -=-; + . . . + i>„ -y-r- ', 



and if n is equal to or exceeds in, we write 



(13) Ws = A„ -r^ + Ai -TTT- c . . . + A„ 



where s = n - m. 



^"dK^^'dK:.' 



d 



dB„' 



It is clear that if we form the eliminant of u + Jcv and v the 

 result must be independent of k; hence we must have E{A,n + IB,,, B„) 

 independent of k. 



Now, the coefficient of k is evidently Q,rB {A„„ B„), and consequently 

 9)1 being greater or equal to n, we must have 



(l-±) a,n.».E{A„„ B„) ^ 0. 



If we now operate with Q,. on the form of B {A,„, B„), given in 

 equation (11), we obtam 



(15) £lrE{A„„B„) = nA„r'BnB„"' + {n - 1) ^„."~'5„X. + {n - 2) A„r'X, 



+ ... + 2A,nB„X„., + A,n"-'arX: + A„r-Qr^2 + A,„"-'QrX, 



+ ... + AMrXn.i + B„X„,, + ... + (- \)"BnE{A,„.u B„) 

 = ^„."-' (?i5„5o'» + QrX,) + A„r- ((« - 1) B„X, + Q,.Zj) 

 + A„r' ( (« - 2) ^„Z= + Q,.X3) + ... + A,,, {2B„X„., 



+ QrX„.:) + B„{X,^, + {-l]"EA,n,,,B„) - 0. 



Now, since Q„E{A„„B,) is identically zero, we are led to the series 

 of equations 



AVi + (-l)"Q,.^(J„,.„^„) = 0, 



2^„2'„.= + Q,X„_, = 0, 



3^„jr„,3 + Q,2'„.2 = 0, 



&c., &c., 



{)i - i)BnX„x, + ax, = 0, 



nB„Bo'" + Q,X\ = ; 



[10^=j 



(16) 



