1887.] Dr T. Muir on the Theory of Determinants. 477 
the second identity may be written 
g h i 
g mx 2 + m x xf + . . . mxy 4- m l x l y 1 + . . . mxz + + . . . 
h mxy 4- m 1 x- [ y l -\- . . . my 2 + ??qy 1 2 +. . . myz + mgjyz +. . . 
i mxz 4- myx.yz^ 4 . . . myz 4 m x y^z x + . . . mz 2 + m x z i 2 +••• I 
g 
X 
x x 
2 
9 
X 
x 2 
2 
g 
x x 
x. 2 
mm 1 
h 
y 
Vi 
4 mm 2 
h 
y 
V2 
+ 
m x m„ 
h 
yi 
2/2 
% 
z 
Z 1 
i 
z 
Z 2 
i 
Z 1 
z 2 
This also is an important theorem, and is not so much an extension 
of previous work as a breaking of fresh ground. 
BINET (November 1811). 
[Sur quelques formules d’algebre, et sur leur application a des 
expressions qui ont rapport aux axes conjugues des corps. 
Nouv. Bull, des Sciences par la Societe Philomatique , ii. 
pp. 389-392.] 
In this paper Binet returns to the consideration of the first of 
the two identities which have just been referred to, writing it now T 
in the form 
~%(xyz" — xzy" 4- yzx ' — yx'z" + zxy” — zy'x ") 2 
= %x 2 Hy 2 %z 2 - %x?{%yz) 2 — '%y 2 (%xz) 2 - %z 2 (%xy) 2 + T%xy%xz%yz . 
He puts it in the same category as the identity 
%{yz - zy) 2 = '%y 2 '2z 2 - ('Syz ) 2 , 
which he speaks of as being then known. Further, he says 
“ Ces deux formules sont du meme genre que la suivante 
f ux'y"z'" - ux'z"y'" 4 uy'z"x - uy'x"z'" + uz'x”y"' 
] 4 xz'tfvJ" - xz'u"y"' 4 xu'z"y'" - xu'y"z’" 4 yz'u"x!" 
[ +yx'z"u r " - yx'u"z'" + zu'y"x'" - zu'x"y'" + zx'y"u"' 
- uz'y"x'" 4 xy'u"z - xy'z"v!" 'j 
- yz'x"u 4 yu'x"z"' - yu'z"x"' j. 
- zxtu"y'" 4 zy'x"u'" - zy'u"x'" j 
2: 
= ^u 2 ^x 2 %y 2 %z 2 — 'Hu 2 %x 2 f$yz) 2 - %u 2 '%y 2 (f£;xz) 2 - %u 2 %z 2 (2xy) 2 
— '%x 2 %y 2 ( K %uz) 2 — ^x 2 %z 2 C%uy) 2 - %y 2 %z 2 {^ux 2 ) 
4- ^%u 2 ^xy%xz%yz + 2 %x 2 ^uy%uz%yz 4 2 %y 2 %ux%uz%xz 
4 2 %z 2 %uxi%uy^xy 4 (%ux) 2 (%yz) 2 4- (^uyfif^xz) 2 + (%uz) 2 (f&xy) 2 
- TLux%xy%y?i%zu — 2 '%uy%yz%zx%x r u ~ 2 '2 i uy%yx%xz%zu , 
