248 Proceedings of Royal Society of Edinburgh. [sess. 
a + 
,2 
a 
+ 
„2 
a 
= 1, 
^ + 
+ 
(3" 2 
= 1, 
0 
y + 
,2 
y 
+ 
y 
= 1, 
f3y + 
Pi 
+ 
P'y 
= cos X , 
ya + 
y’a 
+ 
rr rr 
y a 
= COS fJL , 
a (3 + 
a/3' 
+ 
a "/3" 
— COS V j 
and, since the expression 
L(aX + f3y + yz) 2 + M (ax + (3'y + y'z) 2 + N(a"x + f3"y + y"z) 2 
has to be identical with 
Ax 2 + By 2 + Cz 2 + 2 ayz + 2 bzx + 2 cxy , 
we have thus by implication another set of six equations, viz. : 
La 2 
+ 
Mo.' 2 
+ 
Na" 2 
= A, 
u 2 
+ 
M/3' 2 
+ 
N/3'' 2 
= B, 
Ly 2 
+ 
My' 2 
+ 
Ny" 2 
= c, 
L/?y 
+ 
M/3'y' 
+ 
N/8"y" 
= a , 
Lya 
+ 
My’ a’ 
+ 
±3y"a" 
= b, 
La f3 
+ 
Ma 73' 
+ 
Na "f3" 
= c. 
What, therefore, remains to be done is the solution of these twelve 
equations in the twelve unknowns 
a , P , y : a, /3', y : a", /3", y : L , M , N . 
Jacobi’s mode of accomplishing this is very interesting. He 
notes first that A may be looked upon as known, by reason of the 
fact that it is expressible in terms of X , y , , v , the connection in 
modern notation being 
A 2 
a + a'" + a a/3 + a J3' + a" (3" ay + ay + a"y" 
a/3 + a/3' + a" (3" f3 2 + f3' 2 + /3" 2 y/3 + y'/3' + y"(3" 
. / > , n n rt . n' ’ . n" " 2 \ /2 • " 2 
ay+ay+ay py+ py + p y y +y +y ,, 
1 COS V COS [X 
cos v 1 cos X 
cos y, cos X 1 
In the next place he draws attention to the resemblance between the 
two sets of six equations, and points out that as a consequence any 
equation legitimately obtainable from the second set is matched by 
an equation which might in like manner be obtained from the first 
set, but which is much more readily got by using the substitution 
