72 



Proceedings of the Ro>jal Irish Acadenvj. 



and from these we easily obtain the following 



(17) 





1.2. 



&C., 



&c., 



[i?^,- = (-ir^^^ = ^. 



where h = (- 1)- and E = 2:(^«.„.g,). " 



If we now introduce these values into the value of E {A^ B^, as given 

 in (11), we obtain 



(18) EU..B,) - A^.j^-l;^^^-^S^'-t^'f:S ^ ••• 



+ (- 1)' 



AJ" Q,-F 



hB,c 



E, 



B^ 1.2....;.) 



where E! = E{A^^,B,) and A = (- 1)". 



We have therefore determined E{Am,B.) by the application of the 

 operative process given above, and can write, in general, 



(19) 



E {A., B.) = hB/ ■»• "^ (^.-,. B.). 



(20) E(A^,B.) = hB^ 



we can consequently write 



In a similar manner E {Am^, B.) is expressed by the formula 



^''e{A^,B.). 







■ Cim-m.1 



(21) £(A.,B.) = hB.'c "- e * E(A,.„B.); 



and finally E{A^,B,) is seen to be derivable by an operative process 

 from E {A^ B.). 



We now commence with the eliminant of two quadiatics, and we shall 

 show how all other eliminants may be derived from it. We write 



(22) E{A^ B,) = (^A^,y - (A,B,) (AtB,). 

 where (AiB,) = A^B, - A^,, 



(A^,) = AyB, - A,B^, 

 (A^,) = A,B, - A J!, ; 

 and we propose to find E {!}%, At). 



