122 Proceedings of Royal Society of Edinburgh. [sess.. 
This he holds to be true,* because the product 
f\a n - in )/'(“ n-m + 1 ) • • • • f'(a n ) 
contains as factors the differences of all the elements a 0 , a v . . a n , 
except those which go to make II (a 0 ,a v . . . . a n -m- 1 ) and 
contains a second time but with opposite signs the \m(fn +1) factors 
which go to make n(a w _ w ,a n _ m+ i, . . «w). 
* The factors of a difference-product may always be, and usually are,, 
arranged in the form of a right-angled isosceles triangle : for example, 
£\abcdefg) = ( b - a)(c - a)(d - a)(e - a)(f- a)[g - a) 
(c-b)(d-b)(e-b)(f-b)(g-b) 
( d-c ) (e -c)(f-c)(g-c) 
(e-d)(f-d)(g-d) 
(f~e){g-e) 
( 9-f) • 
Consequently there must be an algebraic identity corresponding toj the- 
geometrical proposition — If from a point in the hypotenuse of an isosceles 
right-angled, triangle straight lines be drawn parallel to the other sides, the- 
triangle is thereby divided into two triangles of the same kind and a rectangle. 
This identity it is which is at the basis of Jacobi’s, for drawing the lines- 
thus — 
(b - a)(c - a)(d - a) 
(c-b)(d-b) 
(d-c) 
(e -a)(f- a)(g - a) 
(e ~b) (f-b)(g-b) 
{e- c) ( f-c)(g-c ) 
(e-d)(f-d)(g-d) 
(f~e)(g~e) 
(9-f), 
we obtain 
Cfabcdefg) = C<abcd) . tf(efg) . (e - a)(f- a)(g - a) 
(e-b)(f-b)(g-b) 
(c-o)(f-c)(g-c) 
(e-d)(f-d)(g-d). 
But the expression here which corresponds to the rectangle in the geometrical’ 
proposition 
= (e-a)(f-a)(g-a) \ 
(e-b)(f-b)(g-b)\ 
(e-c) (f-c) ( g-c ) 
{e-d){f-d)(g-d) }+Ci(efg)-0(9P) 
. (f~e)(g~e) \ 
(e-f) . (9-f) \ 
(e-g)(f-g) • f 
=f , (e)f , (f)f(g) -r (-?(H<f9)-CKef9) • 
Consequently 
meMhfm = (-)rmf)Ag\ 
Q(abcd) 
which is Jacobi’s identity. 
