1899-1900.] Dr Muir on the Theory of Alternants. 
113 
made use of the difference-product, now appears as a theorem 
and with it Jacobi makes his start ; that is to say, he proves 
that 
If in the determinant 
+ aobjCgdg . . . l n _ i 
the suffixes he changed into exponents of powers, the result obtained 
is egual to the product of the Jn(n — 1) differences of a, b, c, . . 1, 
viz., the product 
(b - a)(c - a)(a - a) . . . . (1 - a) 
(c-b)(d-b) (1-b) 
(d-c) (1-c) 
With the help of Sylvester’s notation, which symbolizes the 
opposite change, viz., from exponents of powers to suffixes, this 
may be expressed in the compact form 
£P D(abc . . . 1) = 2 ± af Y c 2 . . ,l n - 1 . 
In proving it he takes for granted (1) that the product in question 
merely changes sign on the interchange of any two of the elements , 
and (2) that in the development of any function of this character 
there can be no term in ivhich two or more exponents are equal, for 
the reason that, if there were one such, there must be another 
exactly like it but of the opposite sign. Combining with this 
latter — which includes of course the case where the index 0 is 
repeated — the fact that, for the particular function under con- 
sideration, the indices must all be + and the sum of them equal 
to \n{n — 1), he concludes that no term can have any other indices 
than 
0 , 1 , 2, . . ., n - 1 . 
Next, as there is only one way of getting an element, k say, in 
the (w-l) th power, viz., by multiplying all the n- 1 binomial 
factors k - a, lc-b, . . . in which k occurs, and after that only 
one way of getting an element, li say, in the (n - 2) th power, viz., 
by taking from out the remaining binomial factors all the n- 2 
factors in which h occurs, and so on, it is inferred that no term 
can have any other coefficient than +1 or -1. Summing up 
VOL. XXIII. 
H 
