of Edinburgh , Session 1885 - 86 . 
583 
&c., les memes relations qu’entre les quantity rj , £, £', &c., 
et a, a, a", /?, &c. (Art. 1), on aura done aussi 
iy'Z" + rjl'h" + Ci'rj" - ££'?/' - - ty? 
= J{aa’a" + 2/3/T/T - a/3 2 - a' /T 2 - a"/?'" 2 ) = A 2 . 
Done on aura cette Equation identique et tr&s remarquable 
iyT + yLT + - tty" - rfC - tv'? 
= (xy'z" 4 - yz'x" + zxty" — xzy" — yx'z' — zy'x") 2 . ” 
The remaining portion is of little importance ; its main contents 
are four sets of nine identities each, viz. : — 
1 . x£-\- x'£ + x'£' = A , 
2. xi+yrj + z£ = A , 
o t _ax + (3"x' + fi'x" 
d. f S 
4> ^ _ a£+b"? +bt 
yi + y'£ + x"£" = 0, &c. 
x'£ + y f r] +z'£ = 0 , &c. 
&c. 
&c. 
Besides the fact that Art. 3 contains a proof of the Lemma of the 
previous memoir, we have to note the new identity 
X = Ax , 
which in modern determinantal notation is 
\\xz"\ \yx"\\ 
\\zx' \ \xy' \\ 
x | xy'z 1 | , 
— a simple special instance of the theorem regarding what is now- 
adays known as “a minor of the determinant ad jugate to another 
determinant.” 
The last two lines of Art. 4 by implication make it almost certain 
that Lagrange did not look upon 
xyd' + yz'x" + zx f y" - xz'y" - yx'z" - zy'x!' 
and aa' a" + 2 bb'b" - ab 2 - a'b' 2 - a"b" 2 
as functions of the same kind. 
The new theorem in Art. 5, which Lagrange justly characterises 
as “ very remarkable,” is in modern determinantal notation 
\y’z"\ 
\z'x"\ 
W 1 
X 
y * 
W 1 
\xz" 1 
1 yx" 1 
= 
x' 
y' 
1 v* 1 
- 1 zx' 1 
\xy' | 
x" 
y" 
