of Edinburgh, Session 1885-86. 
579 
“ Corollaire 2. 
“3. Si on prend les trois equations 
+ y v +z£ = /?, 
xx' + yy' + zz = b" , 
xx" + yy" + zz" = b' , 
et qu’on en tire les valeurs des quantity x, y, z, on aura par 
les formules connues 
T P(y'z" - z'y") + 6 V - if) + b"(fy" - rjz") 
€(y'z” - z’y") + r)(zx" - x'z") + C(xY - y'x") 
P(z'x" - x'z") + b'(tx' - &) + b"(&" - &T) 
V ~ $(y’z " - z'y") + rj(zx" - x'z") + £(a/y". - yx ") ’ 
J3(x'y" - y'x") + b'(jy' - yx') + V’W ~ &') . 
$(y'z" - z’y") + rj(zx" - x’z") + £(x’y" - y’x") ’ 
done faisant les substitutions de l’Art. prec. et supposant pour 
abreger 
a = a! a" — b 2 
on aura 
P( + (a"b" - bb')x' + (a'b' - bb")x" 
x — , 
a 
J3 V + (, a'b " - bb’)y’ + (a'b' - bb")y" 
y j 
a 
/3C + (a"b" - bb’)z’ + (a'b' - bb")z" „ 
a 
In regard to the first identity here (the so-called lemma), tbe impor- 
tant and notable point is that the right-hand member is the same 
kind of function of the nine quantities x 2 + y 2 + z 2 , xx' + yy’ + zz ', 
xx!’ + yy" + zz", xx ' + yy' + zz', x' 2 + y' 2 -1- z' 2 , x'x" + y'y" + z'z", 
xx" + yy" + zz", x'x!' + y'y" + z'z", x!' 2 + y" 2 + z" 2 as the left-hand 
member is of the nine x, y, z, x', y' , z', x", y", z". Indeed, 
without this distinguishing characteristic, the identity would have 
been to us of comparatively little moment. Possibly Lagrange was 
aware of it; but, if so, it is remarkable that he did not draw 
attention to the fact. It is quite true that Lagrange’s identity and 
the modern-looking identity 
x y Z 
2 
x 2 +y 2 +z 2 xx' -\-yy' +zz' xx!' + yy" +zz" 
x! y’ z' 
= 
xx! + yy' + zz' x' 2 + y' 2 + z' 2 x'x" + y'y" + z'z" 
x" y" z" 
xx" + yy" + zz" x'x" + y'y" + z'z" x" 2 + y" 2 + z" 2 
