1896 - 97 .] Dr T. Muir on Ternary Quadrics. 233 
where (7, 1) = A(M^ - np>) + r(bn - cm), 
„ (7,2) = M(Rc -pli) + a(np - Ir), 
„ (7,3)= R(A l ~ bn) + m(pb - qa), 
„ (7,4)= a(mp -lq) + R(cZ-5M), 
,, (7,5)= m(rb - qc) 4- A (Iq-nR), 
,, (7,6) = r(an - cl) + M(^c -pA) . 
This can he directly established by increasing each element of the 
7th row by 
Ir — ISlq times the 
corresponding element of the 1 st row. 
qa — Rc 
5 J 
33 33 2 nd ,, 
cm - A l 
5) 
33 33 3rd ,, 
%R - mp 
3J 
3 3 3 3 4tll 
i 
<1 
3J 
33 33 5tll ,, 
and &M - an 
3) 
33 33 6 th ,, 
when it will be found that the result is 0 in every case. 
(18) The determinant ( 8 ) of § 14 and the similar form of § 17 
may be modified so as to give a form of the eliminant possessing 
all the properties of (8) and at the same time having an additional 
simplicity of form. This modification is effected in the case of 
(8) by diminishing each element of the last row 
by times the corresponding element of the 1 st row 
33 
Rc 
33 
33 
33 
A l 
33 
33 
33 
mp 
33 
33 
33 
rb 
33 
33 
33 
an 
33 
33 
33 
2 nd 
33 
33 
3rd 
33 
33 
4th 
33 
33 
5 th 
33 
33 
6 th 
33 
the result being 
A 
r 
n 
a 
M m 
R 
b 
P 
- M qa - R cm 
l 
b 
n 
a 
R 
- aiiR 
! 1 • V 
A . c 
m M l 
r q 
- mpA ~ rbM. A + A' 
8 
cr 
w 
- Air 
