528 
Proceedings of Roy cd Society of Edhibiirgli. 
The sequence of propositions as might be expected is not unlike 
that found in Vandermonde. The first six propositions are — 
1. The under elements (Aj, A 2 , &c.) being allowed to remain 
unchanged, the upper elements (a^, a 2 , . . . ) are interchanged in 
every possible way to obtain the full development. 
2. The sign preceding each term is dependent upon the number 
of interchanges of elements necessary to arrive at the term. 
3. If two adjacent upper elements be interchanged, the sign of 
the determinant is altered. 
4. If an upper element be moved a number of places to the right 
or left, the sign of the determinant is changed or not according as 
the number of places is odd or even. 
5. If several upper elements change places, the sign of the 
determ inant is altered or not according as the number is odd or even, 
which indicates how many cases there are of an element following 
one which in the original order it preceded. 
6. If in any term the said number of pairs of elements in reversed 
order be even, the sign preceding the term must be positive ; and 
if the number be odd, the sign must be negative. 
The proof of the 3rd of these, which gave trouble to Vandermonde, 
is easily effected in what after all is Vandermonde’s way, viz., by 
showing that the case for n elements follows with the help of the 
definition from the case for 1 elements. (xi. 4.) 
Schweins’ 7th proposition is that there is an alternative recurring 
law of formation in which the under elements play the part of the 
upper elements in the original law, and vice versa. This amounts 
to saying in modern phraseology, that if a determinant has been 
shown to be developable in terms of the elements of a row and 
their complementary minors, it is also developable in terms of the 
elements of a column and their complementary minors. The proof 
is affected by the so-called method of induction, and is interesting 
both on its own account and from the fact that Cauchy’s develop- 
ment in terms of binary products of a row and column turns up in 
the course of it. The character of the proof will be understood by 
the following illustrative example in the modern notation : — 
By the original law of formation we have 
