476 Proceedings of Royal Society of Edinburgh. [july 18, 
<7 2 (BC-F 2 ) +h 2 (AC-E 2 ) + i 2 (AB-D 2 ) 
+ 2^(EF - CD) + 2 gi(m - BE) + 27w(DE - AF) , 
a celle-ci 
%mm\g(yz - zy) + h (zx - xz') + iixy - yx')] 2 .” 
■* 
Now the numerator referred to would at the present day he written. 
A D E 
DBF 
E F C , 
and since %mx 2 , &c. stand for mx 2 -J- m x xf + 7n 2 x 2 2 + . . . , &c., the 
first identity given may be put in the form 
mx 2 + m x x 2 + m 2 x 2 2 + . . 
mxy + m l x 1 y 1 + m 2 x 2 y 2 + . . 
mxz + m-jpcfa + m 2 x 2 z 2 + . . 
mxy + m 1 x 1 y 1 + m 2 x 2 y 2 + . . 
my 2 +m 1 y 1 2 +m 2 y 2 2 + .. 
myz + myy-fo + m 2 y 2 z 2 +. . 
mxz + m^x x z x + m 2 x 2 z 2 + . 
myz + m x y x z x + m 2 y 2 z 2 + . 
mz 2 + m-fi 2 + m 2 z 2 + . 
X 
X x 
x 2 
2 
X 
x x 
x 3 
mmgn 2 
y 
Vi 
V2 
+ mm 1 m 3 
y 
Vi 
y 3 
z 
% 
*2 
z 
h 
% 
+ . . . (xvm. 2.) 
where x v y 2 , . . . are for convenience written instead of x, y" , . . . 
It will be seen that this is an important extension of a theorem of 
Lagrange, the latter theorem being the very special case of the 
present obtained by putting m = m Y — m 2 — 1 , and m 3 = m 4 = . . . = 0, 
— a fact which is brought still more clearly into evidence if, 
instead of the left-hand member of the identity, we write the 
modern contraction for it, viz. 
mx m 1 x 1 m 2 x 2 m 3 x 3 . . . . 
my m l y 1 m 2 y 2 m 3 y 3 . . . . 
mz m x z x m 9 z 2 m 3 z 3 . . . . 
Again the denominator 
X 
x x 
x 2 
x 3 .... 
X 
y 
yi 
y 2 
y% • • • • 
z 
*t 
% 
* 3 .... 
<7 2 (BC - F 2 ) + 7z 2 (AC-E 2 ) + i 2 (AB-D 2 ) 
+ 2y7z(EF - CD) + 2gi(UF - BE) + 27w(DE - AF) 
being in modern notation 
g h i 
g A D E 
li D B F 
i E F C , 
