1899-1900.] Dr Muir on Jacobi's Expansion. 
137 
(b - a) (d - c) . | (ab)° ( cdf | , 
(b - a)(d - c) (/- e) . | (aft) 0 ( cdf (ef)U. 
When, therefore, any one of the sets of linear binomial factors has 
been obtained, we have only to take the product of the elements in 
the first factor, the product of the elements in the second factor, 
and so on : raise these products in order to the 0th, 2nd, 4th, .... 
powers and form an alternant-like permanent having these powers 
for the elements of the diagonal. 
(5) The theorem in its new form thus is : — 
The difference-product of 2n elements may be expressed as an 
aggregate of (2n - 1) (2n - 3) ... 3 . 1 terms , which are obtainable 
by taking the ordinary expansion of the Pfaffian whose elements 
are the n(2n-l) differences arranged in the usual triangular 
fashion , and then annexing to each term of this expansion an 
alternant-like permanent whose diagonal elements are the 0th, 
2nd, 4th, .... powers respectively of the products of the two 
original elements occurring in each of the linear factors of the 
term. 
Or, with a freer use of symbols,. 
The difference-product of a x , a 2 , a 3 ,...., a 2n is equal to , 
+ + 
2(a 2 — a-J (a 4 — a 3 ) (a 6 — a 5 ) .... (a 2 n — a 2 n-i) • I (aja 2 )° (a 3 a 4 ) 2 .... (a 2n -ia 2n ) 2n 2 | 
where (a 2 — a 4 ) (a 4 - a 3 ) .... (a 2n — 2 n-i) i s i n rnagnitude and sign 
a term of 
1 a 2 — a 4 a 3 - a 4 .... a 2n — a^ 
a 3 — a 2 .... a 2n - a 2 
a 2n ~ a 2n-l » 
and where each of the n binary products a 4 a 2 , a 3 a 4 , . . . . a 2n _ 1 a 2n is 
formed from the original elements occurring in one of the linear n 
factors immediately preceding. 
The truth of this may be established as follows: — 
From my theorem for the development of a determinant of the 
(mn) th order * we have, on putting m — 2, the difference-product 
of a x , a 2 , . . . , a 2n , that is to say, the alternant | afaf . . . a 2 ” -1 1 
* Trans. Boy. Soc. Edin., xxxix. pp. 623-628. 
