1896 - 97 .] Dr T. Muir on Ternary Quadrics. 
223 
(or, what is the same thing, multiply the fourth row by A and 
then subtract from it nq - mr times the first row, multiply the 
fifth row by M and then subtract from it pc - ra times the second 
row, multiply the sixth row by R and then subtract from it 
bl - am times the third row) it is at once transformed into 
A(MR — Ip) + a(mr - nq) w(AR - bq) + m(br - A p) q( AM - cn) + r(cm - A l) 
c(MR - Ip) + a(lr - My) M(AR - bq) + m(ar -pc) y>(AM — cn) + r(an - M b) 
&(MR - Ip) + a(mp - R?z) Z(AR - lq) + m(qa - Rc) R(AM - cn) + r(am - bl) 
(y) 
(7) The form of (y) suggests that it is a, product, and with the 
suggestion in mind a little examination suffices to show that it is 
the quasi-product 
A n q mr — nq br - A p cm — Al 
MR -Ip . n 
c M p Ir — My ar - pc an - M6 
9 
AR - bq m 
b l R mp - Rw qa - Rc am - bl 
AM - cn . r 
and therefore by Binet’s theorem can be expanded into 
A n q 
c M p 
b l R 
(MR - Ip) (RA - bq) (AM - cn) + 
A n cm - Al 
c M an - M6 
b l am - bl 
(MR - Ip) (AR - bq)r 
A q 
c p 
b R 
br - A p 
ar -pc 
qa - Rc 
(MR - Ip) (AM - cn)m + 
A br - A p cm - A l 
c ar -pc an - M& 
b qa-~Rc am - bl 
mr(MR - Ip) 
+ 
n q mr - nq 
M p Ir — My 
l R mp - R» 
(AR - bq) (AM - cn)a - 
nmr -nq cm - A l 
M Ir - My an - M6 
l mp - Iln am - bl 
ar(AR - bq) 
+ 
y mr - nq br - A p 
p Ir -My ar—pc 
R mp - R n qa - Rc 
am (AM - cn) + 
mr — nq br - A p cm - Al 
Ir — My ar —pc an — M& 
mp - R n qa - Rc am — bl 
amr. 
0 ?) 
(8) We have thus got two different but closely resembling 
expansions, (/3) and Q3') for our determinant, and when we come 
to compare the two carefully we discover the fact that the one is 
obtainable from the other by interchanging 
a and A , b and c , 
m and M, n and l, 
r and R , p and y , 
