276 
Proceedings of Royal Society of Edinburgh. [sess. 
employing the theorem above called (0). Thus from the first 
column of equations this theorem gives 
Ga = a A + a A' + a"A!‘ 
G a = bA + b A + b A 
G a = cA + c A + c A ; j 
the second column gives similar expressions for G f3 , G'/T , G "13" ; 
and the third column for Gy , G'y , G"y". The whole set is in 
later notation 
Ga G'a G"a 
GjS G'/T G ,f p 
Gy G^y r G r, y r/ 
where 
9, h, h 
a a’ a" 
b b' 
b" 
, „ 
c c c 
) 
aTa /_ a" 
A A' 
A" 
A A' A" 
a a a" 
b V 
b" 
c c c 
B JB' B' 
B B' 
B" 
B B' B" 
a a a" 
b V 
b" 
c c c" 
C C' C" 
C C' 
C" 
C C' C" 
> 
denote gp + h 
cr + Jct 
. Similarly by tak- 
ing the same set of nine equations in wws there is obtained 
( 
Ga G b G ' c 
Ga G'b' G V 
Ga G'b" G 'c" 
a 
p 
y 
a 
P 
y 
a' 
P' 
y 
A 
B 
c 
A 
B 
c 
A 
B 
c 
a 
P 
y 
a 
P 
y 
a' 
P' 
n 
y 
A' 
B' 
G' 
A' 
B' 
C' 
a: 
B' 
C' 
a 
P 
y 
a 
P 
y 
a' 
P' 
y" 
A" 
B" 
G" 
A" 
B /# 
C" 
A" 
B" 
G" 
From these two sets of equations it is clear how the coefficients 
of one of the substitutions may be obtained when the G’s and the 
coefficients of the other substitution have become known. 
Separating the latter of these new sets of nine in a similar 
fashion into column-sets of three, but solving this time in the 
ordinary way, Jacobi obtains a further set, which, if only to save 
space, we may write in the form 
t Jacobi writes the nine equations in one column : they are better arranged 
in three, however. Cayley at a later date would have preferred to write 
more luminously 
ABC 
) ( 
(G , G' , G "\a , a' , a\a , b , c ) 
,t ,e) . . ■ ) 
b b 
— 
(G , G ? , G f, ^a , a' , a"§a f , V , c') 
(G , G' , G"^ , /S' , /3 "\a r ,b' , c') ... 
