(5) 



(7) 



70 Proceedings of the Royal Irish Academy, 



It then appears that u (x, y) can be written in the following forms : 



In like manner we may write 



(6) v{x,y) ^ ifeo'B,, ^ x»e^B,. 



The quantics ?t and v being then completely defined by and derivable from 

 their respective sources, we may denote the eliminant of « and v by tlie 

 symbol E(A„,B„), and write 



{E(A„,B„) = u (£,, jj,) 7t (g„ I,,) . . . « (S„, ,,„), 



/j) . . . i; (3-„, y„). 



Substituting now in the above results for « (?i, i|i) and v{Xi,y), and 

 their values in terms of a-,, yi ; a-j, y, . . . r„„ y„. ; ?,, i), ; li,r\t; ... £„, /u„ ; 

 it is evident, on inspection, that 



(8) E{A,, B„) = (- 1)-" E(B„, A,.,). 



We now write 



E(A^, B.) - \A,.nr + yi™.,.,,-"-'?, + ^„-„„"'-'5,' + . . . + ^„$,"') 



X {yl„n3" + vl„.,.h"-'?j + yl,„-:,i7,"-'53' + . . . + ^o?,") 



X [A^n." + >!„-,»,,-'?,. + A„.tt,„''-'V + . . . + ^o?,,""}, 

 and if we agree to denote the various symmetric functions as follows : 



(9) 



25i ?»•/!. . ij» = 2|,i, 2£i'£3ij2t|)' • • In' = 22,1, &c., &c., 

 we find i-eadily, on expanding the above form of E(A„„B„), 

 (10) E(A^, B.) = A^-B," + A^"-' (.B,--'^„.,2. + B,''^A„,.a, + .. . + ^„2™} 

 + ^."-' (^o'"-'-4'„.,2„i + .B„"-'^„-,^„.,2,,, + . . . + yl/2,„,„.} 



+ An" ' [Bi," A m.i'Sififi + Bq" Am.tA m-\2iltl)l + ■•■ AQiimniDm] 



+ &c., 



4 (-- 1)''B„E(A„.,,B„), since t),»u . . • >j., = .Bg, 

 and $,?,.. „„ = (- 1)»5„. 



It is endent, then, that if we regard the eliminant as a homogeneous function 

 of the .4s of the »i"' order, the coefficient of any term Am-r-Am-fA„.t will 

 be 2r,«, », and consequently if we know the eliminant in terms of the As 

 and the Bs, we can find the symmetric function 2r,„(. 



