558 Proceedings of Royal Society of Edinburgh. [sess. 
adjugate of the determinant of the original set, there is involved 
in Rothe’s proposition the well-known proposition of later times, 
viz., that the adjugate of an axisymmetric determinant is also 
axisymmetric. 
Binet (1811). 
[Memoire sur la theorie des axes conjugues .... Journ. de 
VEcole Polytechnique, ix. (pp. 41-67) pp. 45, 46. 
Sur quelques formules d’algebre, et sur leur application a des 
expressions qui ont rapport aux axes conjugues des corps. 
Nouv. Bull, des Sciences jpar la Societe Philomathique , ii. pp. 
389-392.1 
With Binet we have a recurrence to those axisymmetric deter- 
minants which appear as equivalents to second powers of deter- 
minants or to sums of second powers. His theorems 
mx l + m^ 2 + m 2 xj 2 + . . mxy + m x x x y x + m. 2 x. 2 y . 2 + . . mxz + m x x x z x + m 2 x 2 z 2 + • 
mxy + m x x x y x + m 2 x 2 y 2 + . . my 2 + m x y-j- + m 2 y 2 2 + . . myz + m x y x z x + m 2 y 2 z 2 + . 
mxz + m x x x z x + m 2 x ^ 2 + . . myz + m x y x z x + m 2 y 2 z 2 + . . mz 2 + m x z x 2 + m 2 z 2 2 + . 
2 
+ . . . . 
x x 1 x 2 
2 
ai rv» >y 
vL/ its ^ Vtsg 
= mm 1 m 2 
y Vi y 2 
4- mm 1 m d 
y 9i y 3 
z z x z 2 
"* % 
9 
h ' 
g ma r + mpc.f + 
li mxy + mpc,py Y + 
i mxz + mpcpz x + 
mxy + m 1 x 1 y 1 + 
my 2 + mpjj 2 + 
myz + mpjfa + 
mxz + mpc^ Y + 
myz + + 
mz 2 +m 1 2 1 2 + 
9 
X 
X 1 
2 
9 
X 
x 2 
2 
9 
x i 
x 2 
= mm 1 
h 
y 
Vi 
+ mm 2 
h 
y 
y 2 
+ m x m. 2 
h 
Vi 
y 2 
i 
z 
z i 
i 
z 
Z 2 
i 
z i 
Z 2 
u 2 +w x 2 + 
ux + u 1 x 1 + • • uy + u l y 1 + 
UZ + ttjZj + 
ux + zqaq + 
x 2 ■ 
+ xj 2 
+ ' 
• xy 
+ x 1 y 1 + 
xz 4- 
■ x ih 
+ 
u y + u i9i + 
xy 
+ x iVi + • 
. y 2 
+ 9i 
2 + 
yz + Vtf i 
+ 
uz + u x z 
i + 
xz ■ 
■hx^i 
+ • 
yz 
+ y-fii + 
z 2 -f 
■z 2 
+ 
u 
u i 
u 2 
u 3 
2 
u 
u x 
u 2 
w 4 
2 

X 
'Xi 
x 2 
x s 
+ 
X 
X 1 
x 2 
^4 
4 
y 
y\ 
92 
9z 
y 
9i 
92 
2/4 
z 
z \ 
Z 2 
Z 3 
z 
% 
Z 2 
z \ 
