of Edinburgh, Session 1885-86. 
565 
poses for coefficients a positional notation essentially the same as 
that of Leibnitz, writing ^ where Leibnitz wrote 12 or 1 2 . 
Then he defines a certain class of functions by means of their 
recurrent law of formation — a law and class of functions at once 
seen to be identical with those of Bezout. A special symbolism is 
used for the first time to denote the functions; thus, the expression 
1 0 . %i. 3 2 + 1 1 • 2 2 . 3 0 + 1 2 . 2 0 . 3 X — 1 0 . 2 2 . 3j — lp 2 0 . 3 2 — 1 2 . 2 X . 3 0 , 
which occurs in Leibnitz’s letter, Vandermonde would have denoted 
1 | 2 | 3 
11213’ 
and the result of eliminating x, y , z, w from the set of equations 
l t x + 2 r y + 3 r y + 4 r w = 0 (r = 1, 2, 3, 4) 
by 
1 I g I 3 | 4 . 
1 | 2 | 3 | 4 
It is next pointed out that permutation of the under row of 
indices produces the same result as permutation of the upper row, 
that the number of terms is the same as the number of permutations 
of either row of indices, and that half of the terms are positive and 
half negative. 
The part which follows this is a little curious. The proposition 
is brought forward that if in the symbolism for one of the functions 
a transposition of indices takes place in either row, the same 
function is still denoted, the only change thereby possible being a 
change of sign. The demonstration is affirmed to be dependent on 
two theorems, neither of which is proved, as the proofs are said to 
be troublesome to set forth. Now it will be seen that the second 
of these theorems is to the effect that the transposition of any two 
consecutive indices causes a change of sign, and that consequently 
this alone is sufficient for the required demonstration. The first of 
the auxiliary theorems, in fact, is an immediate deduction from the 
second, the particular permutation which it concerns being produced 
by ( n-m+ 1 )(m- 1) transpositions of pairs of consecutive indices. 
Passing over the illustrations of these propositions, we come next 
to the theorem that if any two indices of either row be equal the 
