154 Proceedings of the Royal Society of Edinburgh. [Sess. 
1 
X 
y 
z 
= 
X 
y 
z 
x^y^z 1 
X 
z 2 xy 
y 2 xz 
X 
z 
y 
y 
z 2 xy 
x 2 yz 
y 
z 
X 
z 
y 2 xz 
x 2 yz 
z 
y 
X 
Similarly from the equation x-\-y + z-\-w — 0 or any one of its seven 
relatives by using the multipliers 
1 , zw , ytv, xiv , zy , zx , yx , xyzw 
and eliminating 
xyz , xyw , xzw , yzw , x , y , z, w 
there is obtained the result 
. 1 
1 
1 
1 
. 1 
1 . 
IV 2 
z 2 
1 . 
1 . 
w 2 
y 2 
1 1 
. w 2 
a ? 2 
1 
1 . 
z 2 
y 2 
1 
. 1 
. z 2 
X 2 
1 
1 . 
• y 2 
X 2 
w 2 
z 2 y 2 
X 2 . 
+ y + ^ + w)(x -y + z + w)(x + y 
• ( -x- y + z + w)( - x 4 
[ g 
stated to be the axisymm 
X 
y 
z 
w 
. X 
y • 
IV 
z 
X 
z 
w 
y 
y ■ * 
w 
X 
X 
w 
z 
y 
y 
IV 
z 
X 
z 
w 
• y 
X 
IV 
z y 
X 
The general theorem is then formulated as follows: “Si en general on 
considere n elements x ly x 2 , . . . , x n , en posant 
X n = + . . . + X n , 
et en designant par 
|X n (l, 2, . . . , m)\ 
le produit des facteurs lineaires qu’on deduit de X n en changeant les signes 
a m des elements x x , x 2 , . . . , x n ; en aura pour n impair 
X„* I X n (l) | • | X n (l , 2) | ... |X n (l, 2, 3, . . . ,K"-1)I = - A ’ 
