1901-2.] 
Dr Muir on the Theory of Orthoyonants. 
277 
A a 
A a 
A a" 
( 
| aB’C" | 
| &B'C" | . 
) 
j cB'C" j 
TT 
G' ' 
Al 
G 
M 
G 
A/T 
G" 
= 
| aC'A" | 
| hC'A" | 
| cC'A" | 
Ay 
G 
Ay 
G 
A/ 
G" 
| «A'B" | 
| bAB" | 
| cAB" | 
where A = | AB'C" | , or, as Jacobi of course writes it, 
A = A (B'C" - B"C') + B(C'A" - C"A') + C(A'B" - A"B') . 
From a set giving the Italic coefficients in terms of the Greek 
coefficients we have thus got a reverse set. The other reverse set 
obtainable in the same way need not be given; but it is easily 
seen that the two have the same practical value as the two from 
which they are derived. 
To make another advance, either of our latest sets of nine is 
taken and separated into row-sets of three, and theorem (0) applied. 
The result which Jacobi gives in nine separate equations of the 
type 
B'C" - B"C' _ a a a b aTc_ 
A G G' G" 
may be written more compactly and more instructively in the 
form 
| B'C" | 
| C'A" | 
| A'B" | 
1 ( 
^ aCL a'b 
a"c 
+ G" 
Ba 
£'& . y3 "c 
y a ,y'b ,y" G 
G G' ^G" 
A 
A 
A 
G + G' 
G 
+ G r + Q ir 
| B"C | 
| C"A | 
1 A"B I 
acd a'b' 
G G' 
a"c' 
_i 
&a' 
j8 'b' /3 V 
yd' y'b' y V 
A 
A 
A 
+ G" 
gT 
+ G , G „ 
G G' ^ G" 
|BC' 1 
1 CA' | 
| A"B' | 
acd' a'b " 
[ G G' 
a"a" 
Ba" 
’ B'b" B"d' 
ya" y'b" y'c" 
A 
A 
A 
+ G" 
Gr 
G' + G" 
G G' G" 
Any one of the nine here, however, may be matched by one 
deduced directly from the set of nine which we obtained at the 
very outset. Thus* 
* Nowadays we should rather put 
| B'C" | — I G/3cd + G'/3'b' + G"j8"c' Gfial' + G A'b" + G" f$”c" \ 
I G yo! + G'y'b' + G"y"c' GyCt" + G'y'b" + G"y"ti" , 
I G/3 G'/3' G"J3" I i a! b' c' I 
I Gy Gy G"y" I I a" b" c" \ , 
— GG' ■ | fiy' \ ’ | a'b" | + GG" * | I3y" | * | a' d' | + G'G" * | Ay" \ * [ b'd’ | . 
