1894 - 95 .] Dr Muir on ProUem of Sylvester's in Elimination. 373 
5. From (a) we first obtain the set corresponding to Sylvester’s 
derived set for the simpler problem, viz., 
2CQA'.t 2 +(ABC + PQK)yz- 2BCC'zo;- 2QBB'zy = 0 
2AEBy - 2RPCV^ + (ABC + PQR>a;- 2CAA'xy = 0 
2BPC'z2 - 2km yz - 2PQ A’^^ + (ABC + T(iU)xy = 0 
the mode of derivation being indicated by the operational formula 
PB^2(ai) - PQa;2(a2) - BAy2(a3) = - xy(/3 ^) . 
6. Another similar set is got by solving (a) for x'^, z^ in terms 
of yz^ zXf xy, viz.. 
(ABC + PQR)z2 = 2RPC'icy + 2 ABB'^a; - 2APAy ' 
(ABC + PQR)y2 = 2BCC'xy - 2BqB'zx + 2PQA'y^ - 
(ABC + PQR)22 = _ 2GBC'xy + 2QRB'zx + 2 CAAV 2 ; 
• • (y) 
7. Substituting in (/3) for xy, yz, zx by means of (a) we obtain 
three equations in y’^, viz.. 
2A'(2CQA'B'C' - BC^C'^ - B'^Q,m)x^ 
+ B'{(ABC + PQR)RC' - 2AQRA'B'}y2 
+ C'{(ABC + PQR)BB' - 2BCPA'C'}22 = 0 . . . (8) 
and two others. 
8. From (a), as we have seen, x'^ can be expressed in terms of 
yz, zx, xy, so that substitution in (^j) gives us a set in yz, zx, xy, 
viz., 
2QR{(ABC + PQR)B'-2CPA'C' }xy 
+ {4ACPQA'2 - (ABC + PQR)2}yz 
+ 2BC{(ABC + PQR)C'-2AQA'B'}saj = 0 . . (c) 
9. Proceeding in a converse way, viz., using (^^) and to give 
us and 2 ^, each expressed in terms of yz, zx, xy, and then substi- 
