256 Proceedings of Royal Society of Edinburgh. [sess. 
and solved as sets of linear equations, the results being put in the 
form 
k(Aa 
Mb' 
+ 
MP' 
+ 
M’P") 
= 
G^a , 
k(Ba 
+ 
B 'b' 
+ 
B "b" 
+ 
B "P") 
= 
cp. 
k(C a 
+ 
CP 
+ 
C"b" 
+ 
CP") 
= 
G"'y , 
k(Da 
+ 
BP 
+ 
D "b" 
+ 
D"P") 
= 
G8 ) 
k(A V 
+ 
Ma 
+ 
MP" 
+ 
M"c ") 
S 
G a , 
k(Bb' 
+ 
B 'a 
+ 
B "c" 
+ 
B"'c") 
= 
cp, 
k(Cb' 
+ 
C'a! 
+ 
Cc'" 
+ 
C'"c") 
= 
G"V , 
k(Db‘ 
+ 
D 'a 
+ 
D "c" 
+ 
D '"c") 
= 
G8'; 
k(Ab" 
+ 
Me" 
+ 
A" a" 
+ 
M"c) 
= 
C'a' , 
k(Bb " 
+ 
B P" 
+ 
B "a" 
+ 
B'"c) 
= 
G"/r , 
k(C b" 
+ 
CP" 
+ 
C "a" 
+ 
C'c) 
C'"y" , 
k{Db" 
+ 
DP" 
+ 
D "a" 
+ 
D"'c) 
= 
GS"; 
k(Ab'" 
+ 
Me" 
+ 
MP 
+ 
M"a") 
= 
CP" , 
k(Bb'" 
+ 
BP' 
+ 
B V 
+ 
B"'a") 
= 
C"fi" , 
k(Cb"' 
+ 
C'c" 
+ 
CP 
+ 
C'a") 
= 
C'"y" , 
k(Db'" 
+ 
D'c" 
+ 
D'P 
+ 
D'"a") 
= 
G8'" ; 
where it is readily seen what is denoted by A , B , C , D , A' , B' , 
... * The corresponding results from the other set of ten 
equations are 
a = 
-kA 
p = 
-kB 
y = 
- kC 
8 = 
kD, 
a = 
kM 
p = 
kB' 
/ 
7 = 
kC 
8' = 
-kD', 
a" = 
kA" 
p = 
kB" 
y = 
kC" 
8" = 
-kD", 
m 
a = 
kM" 
P" = 
kB'" 
nt 
7 = 
kC" 
S'" = 
- kD"' , 
these being most quickly obtainable by means of the specialising 
substitution just referred to. By taking each result of the former 
* Observe A is not the co factor of a , viz., | py"d"' \ , but 
| 0 V'8'" | -f | a/ 3' 7 "5"' | . 
I have drawn attention elsewhere to the fact that at this point a passage 
occurs which contains Jacobi’s first printed reference to determinants. The 
words are “ Valores sedecirn quantitatum A , B , . . supprimimus eorum pro- 
lixitatis causa ; in libris algebraicis passim traduntur, et algorithmus, cuius 
ope formantur, hodie abunde notus est.” 
