ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 



669 



(6) The first step necessary for the purpose in view is to go through the work of 

 the two methods — Sylvester's and Salmon's — as applied to a triad of perfectly general 

 ternary quadrics, so as to have the resulting eliminants actually before us. 



Let then 



x y 



z 



ffi A 



A 



x y 



z 



a 2 h 2 

 9i A 



9 2 



A 



C 2 



ll„ = 



X 



y 



z 



H 



h 



9 3 



h 3 



h 



A 



9z 



A 



C 3 



so that 



J(u v u 2 , w 3 ) = 



a 3 x+h 3 y+g 3 z 



\x+\y+f Y z 

 h 2 x + b 2 y+f 2 z 

 h 3 x + b 3 y+f 3 z 



9i x +Av + c i z 



9z X + f29 + C 2 Z 



9 'z x +/0+ c s s 



+ {l*A/ 8 | + l*MI}afy + {I^APsI + KVsI}^ 2 



+ { I V 2 c s 1 + 1 gAA 1 }f z + {1 9 Ah I + 1 A i/ 2 C 3 1 }y* 2 

 + { I «i/ 2 c 3 1 + 1 9 Ah I Y~ x + {I a Ah I + ! a iA9s I } zx * 



+ {KVsl + 2 \A9ih I } X V Z > 



and 



— = Z\a 1 h 2 g 3 \ 



X<1 + {iWsl + IVV/slh/ 2 + {KMI + lftMl}** 



+ {\a 1 b 2 c 3 \ + 2\f 1 g 2 h s \}.yz + 2{\a 1 f 2 g s \ + \a 1 h 2 c 3 \} ■ zx 



+ 2{\a 1 b 2 g 3 \ + \a 1 h 2 f 3 \}-xy. 



Following Sylvester's method we thus have the eliminant in the form 



«1 



. 



2\ 





flj 6 X 





2 9i 



2/x 



h 



. 



a i 



2/x 



2Aj 



c i 



• 



2^ 



. 



c i 



. 



\ 



tyi 



2/i 



a x 



2A X 



a 2 



. 



2h 2 



. 



^2 & 2 





2^2 



2/2 



h 



. 



<h ■ 



2/ 2 



2A 2 



C 2 





2^2 



. 



h 



. 



\ 



2^ 2 



2/2 



a 2 



2A 2 



a 3 



. 



2h 3 





C 3 & 3 





2^3 



2/3 



h 



. 



a 3 



2/3 



2^ 



h 



• 



25-3 



• 



«i 





&3 



2^3 



2/3 



a 3 



2h 3 



Ml IW 3 I 



9iAh\ + 



«i Vs 1 

 a A9s\ 



+ lV 2 C 3l 



\9AA 1 



_J«i/AI .I^A&I 



\9\Kh\ 1 a A/a 1 



l^iVsl + 



1 ^l/ 2 C 3 1 



«A C 3 I 

 ^1/2^3 1 



+ \a 1 b 2 c s \ 

 1\A9%\\ 



and following Salmon 



s method we have it in the form 









a x 



\ 





c i 



2A 



2 9i 





2\ 



«2 



\ 





C 2 



2/2 



2^2 





2h 2 



a z 



h 





C 3 



2/3 



2^3 





2h 3 



3 l a Ml 



Wg/sl+I^MI h 



Ah\ + ^A C s\ 



\aJ> 2 c 3 \-\-2\f x g 2 li 3 \ 2\a 



-iA9z\+ 2 \ a A4 



2K&25 , 3l + 2l a A/ 3 



"iVJ+KMI 



3IW.I 



to 



6 2 C 3 l + l^l/2 C 3l 



2\KJ> 2 c 3 \ + %6 2 / 3 | |«MI + ai/i^Al 



2|V 2 . 



7 3 | + 2|aA/ 8 



\ h &\+KAg 3 \ 



MaI+IsA 



u 



Sl^i/gCjl 



21^xV 3 l + %/Al 2 k 



hA c s\ + 2 \9A c s\ 



\ a A 



C 3l + 2 l/l^3 



