1896 - 97 .] Dr T. Muir on Ternary Quadrics. 227 
These matters being premised, the expansion of the determinant 
is seen to be 
o 
(o- - PiXo- - p 2 )(<r - p 3 ) + + a i a 2 a 3 “ ^a 1 A(cr - p 3 ) 
or 
© o o 
O- 3 - O- 2 ^! + (T.SpjP 2 - p 2 p 2 p 3 + AAA + a i^o a 3 - ^«l/5 2 (o" “ P 3 ). 
But the coefficient of cr 2 is clearly 
- 22A.y>.Z ; 
and the coefficient of cr is 
© O 
^P)P‘2 ~ ^ a lA 
which, after actual multiplication, is found to be 
.pb.lq. + 22A .m.p 2 .cl 
- 32AM .am.p.q - 2(AMR2A.^.Z) + %(2amr'%A.p.l) 
+ 222AM .pb.lq - 22A.ra.y? 2 .cZ - 3 bnp.clq. 
Writing the former term as a term in a, viz. 
cr(AMR + amr)( - 22 A.p.l ) 
o O O 0 
or <r( - AMR2A.y>.7 - AMR^a.g.« - amr'ZA.p.l - amr%a.q.n ) 
or <r{ - 2(AMR.2A.p.Z) - 2(amr2A.y>.Z)} , 
we see that the terms in cr 2 and <x readily combine, with the result 
The terms independent of cr are 
~ P1P2P3 T a i a o a 3 + A A A} ~ ^ a iAP3 
o 
or - P1P2P3 + Sa^jjag - 2 a 1 A 2 p 3 , 
and it is found that 
© 
- p 1 p 2 p 3 = - cr. bnp.clq - %'%AAL.r.pb 2 .l 2 q , 
that 
2 a 3 = (Zmy> + cZy) (A 2 M 2 R 2 + a 2 m 2 r 2 - 4 AMR.amr) 
+ (Zwp + clq)^^A.p.l(amr - AMR) 
+ ( bnp + clq^^A^i.pb.lq 
- (bnp + clq)bnp.clq 
o 
- (bnp + clq)% AAi.am.p.q 
+ 2(2AMR - amr^A.am.np 2 
+ 2(2 AMR - amr)%A.ra.l 2 q 
o 
- %bnp%A.m.p 2 .cl 
- 22A 2 M.am 2 .p 3 + 22AM 2 .a.6.Z2 3 + 22A 2 AL.m.p%,l - 222AM. mr.p 2 b 2 , l 
