289 
1909-10.] Dr Muir on the Theory of Orthogonants. 
The next step is the deduction of an important result from the considera- 
tion of the determinant 
\f l(*2> • ■ • !/.(*») 
if A* i) • • • iAK) 
J/»- l(*l) i./Vlfc) • • • ifn-l{x n ) 
or A say. 
Each element being linear in the x’s, the determinant is of the n ih degree 
in those variables, and therefore 
= 2 B 
ei'2 • 
rvfi l r e 2 
On the other hand, if we substitute for each element its expression in 
terms of the % s, the result is manifestly a product determinant, and we 
learn that 
j an 
a 12 
a 13 • 
• • a iM 
g& 
g \£ i • 
• • ffr'fi 
&21 
a 22 
a 23 • 
• • «2 n 
• 
& 
9^2 
0 % • 
• ■ ar% 
1 a nl 
a n2 
a n3 • 
• • a nn 
QrJan 
9rJ=n • ■ 
. • gT l L 
— ( ± 1 ) • I Q\q\ • • • Qn 1 I * • • • in • 
Equating these two values and substituting the expression found at the 
outset for . . . £ n we obtain 
2*. 
ei«2 • . • e n 
x^x? 
. x: 
= (±i) 
ahl ■ 
tir 1 
2 A 
eie 2 
e 2 
tA./ 2 • 
. x: 
and thus see that 
as Jacobi had shown in 1845 in the case of n = 3. With the help of this, 
Hesse’s first result at once becomes 
and the desired result is reached. 
B 2 
e\e 2 . . . en > 
[List of Authors. 
