114 
Proceedings of Royal Society of Edinburgh. 
rather hurriedly, he consequently finds that the development of 
the product may be got by permuting in every possible way the 
indices of the term 
a°b l c 2 . . . Z n_1 
and determining the signs in accordance with the law that the 
interchange of any pair causes the aggregate of all the terms to 
pass into the opposite value. This being exactly the mode of 
formation of the determinant %±a 0 b 1 c 2 . . . l n -i with the differ- 
ence that suffixes take the place of exponents of powers, the 
theorem is held to be established signis insuper ea lege 
definitis ut binorum indicum commutatione Aggregatum omnium 
terminorum in valorem oppositum abeat. Quse ipsa est Determin- 
antis formatio, siquidem exponentes pro indicibus habentur ”). 
In passing, he remarks on the large number of vanishing terms 
in the development of the product, viz., 2* w ( n " 1 ) — n ! , and the 
consequent desirability of obtaining this development from that of 
the determinant and not vice versa. 
The fundamental relation between the determinant % ± a 0 b 1 c 2 ...l n - 1 
and the product of the differences of a, b, c, . . ., I having been 
established, it is then sought to find properties of the latter from 
the known properties of the former. What properties of the 
determinant are used Jacobi does not mention, all that is given 
being a bare enunciation of the results. It may be as well, how- 
ever, to point out at once that all of them flow from one general 
theorem, viz., that of Laplace regarding the expansion of a 
determinant in terms of products of its minors. 
The first is indicated by using as examples the case of three 
elements, a v a 2 , a 3 , and the case of four elements, a lf a 2 , a 3 , a 4 , 
viz., 
(a 2 - a 1 )(a 3 - cq)(a 3 - a 2 ) = a 2 a 3 (a 3 - a 2 ) 
+ - a 3 ) 
+ a 1 a 2 (a 2 -a 1 ) i 
(a 2 - a^){a 3 -a x ) (a 4 -a s ) = a 2 a 3 afa 3 - a 2 )(a 4 - a 2 )(a 4 - a 3 ) 
- a 3 a 4 afa 4 - a 3 )(a Y - a 3 )(« 1 - a 4 ) 
+ a 4 a l a 2 (a l - a 4 )(a 2 - a 4 )(a 2 - cq) 
— a^a 2 a 3 (a 2 a^)(a 3 — flq )($ 3 — a 2 ), 
