282 
Proceedings of Poycd Society of Edinburgh. [sess. 
whence it follows that the determinant of either matrix is equal 
to | AB'C" | 2 , and the secondary minors equal to 
( I B'C" 
| B'C"| | A'B"| + ... 
| C'A" 1 1 A'B" | + . . . 
i a'b" i 2 + ... 
( , 
) ( 
l 
n' m' 
n' 
m V 
m! 
l' n 
i 2 + 1 B"C I 2 + 1 BC'| 2 | B'C" 1 1 C'A" I + 1 B"C 1 1 C"A \+\ BC' 1 1 C A' | 
| C'A" I 2 + | C"A | 2 + ICA'I 2 
Then from the original set of nine equations we have 
G a + G' a'~-fG" a" G a/8 + G' a'jS' + G" a' 3" G ay + G ay + G"a'y"' 
G~0~ + G' 3' +G 3"^ G ^y + G' 3'y* ^G" 2 3" y" 
•2 2 2 2 ,2 2 
G y + G y + G" y 
and from this, in passing, by the addition of diagonal elements, 
2 2 2 
l 4- m + n = G + G r + Cl ' . 
Next, as the matrix on the right 
( . ) ( 
there follows 
G a Gc'^a' G” a 
a P y 
Gr 2 ft G ,2 d' G" 2 /T 
2 - .9 ..9 .. 
a p y 
Gy Gy Gy 
a" /?" y" 
< , - . ' , ) ( 
) ( 
n ) 
G a G a G a 
l n m 
a a a 
G 2 £ G’ 2 P G' 2 /3" 
' = 
n m l' 
ft ft' ft" 
G 2 7 G'V G'V 
m l n 
y y' y" 
^ la + nft + m'y la + nft' + m'y la" + n ft" + my" 
na+mft + l'y na! +mft' + l'y na" + mft" + ly" 
m a + lift + ny m a + V ft' + ny m a" + V ft" + ny" 
whence, by summing the squares of the elements of each row 
separately, we have 
/N „ 2 ,2 ,-,,4 ,-/2 7 2 ,2 ,2 \ 
G a + G a + G a = / + n -J- Wi , j 
r^n o 2 '4 n r 2 a /; 4 n" 2 2 . 7' 2 , '2 L 
up + 6 p + G p = m + / + , j 
u y + G y + G y — n + m + Z 
Among the results obtained up to this point, there are sufficient 
to determine the twenty-one unknowns, and to this Jacobi now 
definitely devotes a section (§ 14). First the G’s are dealt with. 
There having been obtained 
