ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 



35 



000 



+ 2012 



+ (123-456) 







5 



8 + 



9 + 9' 







6 



4 



7 + 7' 











5 



9 



8 + 8' 



IT 



6 



6 



8 



12 



7 



II 



6 



4 



8 



12 



10 



5 



9 



+ 2ooi 



- 2015 







6 



8 



7 + 7' 







5 



5 



8 + 8' 



IT 







12 



8 



7 + 7' 



9 



5 



5 



6 



II 



2 004 



+ 2 045 







4 



5 



8 + 8' 







IT 



6 



9 + 9' 



8 







11 



6 



9 + 9' 



8 



12 



4 



5 



9 



Now of the seven complex terms here it is seen that three have 00 for a factor, that is 

 a factor of other three, and that in the last does not appear at all. A little investiga- 

 tion, however, suffices to show that 00 is a factor of every one. Thus, taking the first 

 of the second three, we have 



2012 







5 



9 



8 + 8' 



II 



6 



6 



8 



12 



and similarly 















12 



8 



-2oi5 



7 + 7' 



9 



5 





5 



6 



II 



and 







IT 



6 



2 045 



9 + 9' 



8 



12 





4 



5 



9 



2 012{0(11 12-68) - (8 + 8')(512-89) + 6(56-911)} 



2012{0'03 - (8 + 8+06' + 6-08'}, 

 20° 12 (03-6'8-6'8' + 68'): 







- -20015(56'-37-37'-08'), 



20045(34+8'9 + 8'9'-06'), 



The seventh and last, which is the most interesting, may be dealt with in the same way, 

 but the resulting identity, viz. 



7 



II 



6 



4 



8 



12 



10 



5 



9 



000', 



is a known theorem regarding compound determinants.* 



* See Mum's " Determinants," p. 216, ex. 7. A more general theorem is obtained thus 

 ' a x lj 3 1 | \m t f 3 1 I Cl n.J 3 



a ih93 I I h m 293 I I c i>h93 

 ajl 2 h 3 1 I bjm^ | | c x nji 3 



= I a lkf3 \{ h m 293 I I Wis I - I h m A I I C l n 293 1} 

 " I <h l &3 l{l & l m 2/s I I <V»g&3 I - I \ m i h 3 I I ^2/3 1} 



+ I ajohs \{\ 6 1 m 2 / 3 | | c v n$ 3 \ - \ b^m^ \ \ c x nj 3 |} 

 and this expanded form may, by the use of the theorem 



I ^2^3 1 I c i^3 1 = I l x m 2 c 3 1 I 9l n 2 h 3 I + I & 1 m 2 7z 3 1 | c^^ \ + | \m 2 K 3 \ \ c 1 n 2 g 3 \ 



