225 
1896 - 97 .] Dr T. Muir on Ternary Quadrics. 
(10) The fact that the interchange 
( a m r b 
1 
AM R c 
n p 
b g< 
leaves the eliminant unaltered, gives us at once an alternative form 
for (a), viz., 
a 
m 
r 
A 
n 
9 
c b 
M l 
p R 
Ip - MR . . mr — nq rl — Mg mp — Rw 
. qb - RA . rb - Ap ra-pc aq -Re 
. . cn — AM me — A l an - M5 am — bl 
(«) 
(a) and (a) being related to each other as (} 3 ) and (/3') are. This 
does not occur, however, in the case of (y), because the like sub- 
stitutions made there merely change rows of the determinant into 
columns. The form (y) thus has an advantage over (a) and (/?) in 
that it makes evident the symmetry which is known to exist. 
To prove that (a) and (a) are identical is a good problem in the 
transformation of determinants. 
(11) The symmetry of the eliminant, with respect to the said 
interchange, can also be utilised in finding the final expansion of 
the eliminant ; because, if any one term of the expansion be got, 
another can at once be written down by substitution, unless, of 
course, the term happens to be invariant to the substitution. 
The existence of the cycles may be made use of for the same 
purpose, as any term not symmetrical with respect to the cyclical 
substitutions will in this way give rise to two others. 
Ordinarily, therefore, one term when found suffices to determine 
five others, so that the terms may be arranged in sets of six. Thus, 
if it be ascertained in any way that 
bnp.clq.A.p.l 
is a term of the expansion, we at once conclude that it is only one 
of three, viz., 
bnp.clq.A.p.l , bnp.clqM.b.q , bnp.clq.'K.n.c 
VOL. XXI. 21/12/96 P 
