1901-2.] Dr Muir on the Theory of Orthogonants. 
281 
The reason for this dualism is at once perceived on noting that 
the original set of nine equations is matched by a derived set, 
perfectly similar in form, but having (B'C" - B"C')/' A , .... in 
place of A, ... . Of course, as Jacobi notes, the dualism 
extends to the transformation which is the object of the whole 
memoir ; that is to say, the equation 
(A^ + By + C z)w + (A'x + By + Q!z)w + (A"x + B"y + C"z)w" 
= G st + G'su + GW 
is necessarily accompanied by 
[(B'C"-B"C> + (C'A"-C"A '),j + (A'B"-A"B» 
+ [(BTC - BC" )x + (C"A -CA ")y + (A"B - AB" )?J\io 
+ [(BC' - B'C )x + (CA' - C'A )y + (AB' - A'B )z]w" 
— G G st + G "Ga'u + GGV'y . 
This means, in modern phraseology and nomenclature, that the 
linear orthogonal substitutions lohich cha7ige 
X 
y 
z 
A 
B 
C 
w 
A' 
B' 
a 
w 
A'' 
B" 
C" 
w 
(jst + G ' su + G'tf'v 
will at the same time change 
x y z 
| B'C" | | C'A" | | A'B" | 
| B"C | j C"A | | A"B | 
| BC' | | CA' | | AB' | 
w 
w 
into 
G'G "st + G"Gs u + GG ^ v . 
In parallel columns with these results regarding the p’s and g’s 
Jacobi places a series of others perfectly similar to them, the twin 
series originating in the fact that in squaring | ABC" | , as we 
should nowadays put it, the multiplication may be performed 
either row-wise or column-wise. The chief points in the second 
series we may state rapidly in modern compact form as follows. 
By way of defining the new letters introduced we start with 
l n m 
n m V 
ml'n 
. ) 
A 2 + A'% A" 2 AB + A'B' + A"B" AC + A'C' + A'C" 
BA + B'A' + B"A" B 2 + B' 2 + B' 2 BC + B'C' + B"C' 
CA + C'A' + C"A" CB + C'B' + C"B" C 2 + C' 2 + C" 2 
