1901 - 2 .] Dr Muir on the Theory of Orthogonants. 
283 
9 9 9 
G + G' + G" 
_ ,9 ..9 o o 2 2 
G G + G 2 G 2 + 6 6 
= l + m + n = p + p +p" , 
2 , fr 2 
= ( mn - V ) + ■ • • • = {pp -q) 
GG'G = A 
+ • • * 
2 2 2 
it is perceived at once that G] G' , G"“ are the roots of the equation 
39 2 2 ,2 
# - af (Z 4- m + rc) + x{mn + nl + Im - V - m ' - n ) 
, , ,2 ,2 ,2 
- (/ 77 m + Tim n -ll - - nn ) = 0 , 
or 
3 2 t n tun 2 ,2 |,2 
x - x (p+p +p') + x(p'p" +p"p + pp -q — q — q ) 
n f ft 2 ,2 9 ,/2 
- (j)p p" + 2qq q" - pq -pq -p q = 0; 
which respectively are the same as 
2 2 2 
(a? — — m)(a? — n) — l' (x — l) — m {x — m) — n (x-n) - Tl mn — 0 , 
2 / 2 /r 2 tf f n 
(x -p)(x —p)(x - p") —q{x -p) - q (x - p) - q (x -p") - 2 qqq" = 0 ; 
and either of which is 
^3 _ a ; 2( A 2 + B 2 + Q 2 + A '2 + B '2 + Q ’2 + A "2 + B "2 + Q - 2 ) 
( ( B'C" - B"C') 2 + (O' A" - C"A') 2 + ( A'B" - A"B') 2 ] 
+ xl +(B"C -BC") 2 + (C"A -CA") 2 + (A"B -AB") 2 V 
( + (BC' -B'C ) 2 + (CA ; - C'A ) 2 + (AB' -A'B ) 2 j 
- { A(B'C" - B"C') + B(C'A" - C"A') + C(A'B" - A"B')} 2 = 0 . 
As an alternative to this, however, it is pointed out that we might, 
by putting the equations 
G 2 a = la + n /3 + m'y 
G 2 /3 = n'a + m/3 + l'y 
G 2 y = m 2 a + Z'/3 +ny 
f 0 = (l — G 2 )a H- n’p+ m'y 
in the form J 0 = n'a + (m - G 2 )/3 + Ty 
[0 = m a + l f$ + {n — G 2 )y f 
eliminate a, /3 , y and obtain a cubic in G 2 ; then by similar action 
,2 ,,2 
obtain the same cubic in G and the same cubic in G . In this 
22 
way the left-hand side of the equation, whose roots are G , G'", 
o 
G" J , would naturally recall determinants, although Jacobi does not 
say so; and after Cayley (1841) it might have been written 
t x n m 
p - x q" q 
n m — x l' 
or 
q" p — x q 
m V n — x 
q q p" - x 
