381 
1905-6. 
Dr Muir on the Theory of Alternants. 
The result of the whole change was therefore 
(-ir 
1 = £ 
A (a 1 , a 2 , . . . , a n ) ’ 
whence it followed that 
£ = ( — 1) ? A(oq } a 2 f ... , a n ) ) 
and so the longed-for result was reached 
J = (- ■ ■ ■ .“.+!> [! ,., n + l \ 
AjA.j . . . A„ 
Thereupon additional results come with a rush. First we are 
told that in a similar manner the determinant got from J by 
changing the second power in the denominator of every element 
into the first power * is found equal to 
/ _ ^(^1 ’ ^2 ’ ‘ ' • ’ ^n) • ^( u l ; U 2 > • • • > ^n+l) 
V ' A 1 A 2 . . . A n 
Then “ E combinatione aequationum prod it 
det. j 
f 1 
1 
1 1 1 
! (a 1 + uf 
’.(« 2 +“) 2 ’ ' 
( a n + u ) 2 ’ J 
det. 1 
f J 
1 
1 i [ 
i + u 
5 ? * 
a 2 + u 
a n + u ) 
=[i. 
+ .!]■ 
u=u 1 , =u 2> = 
: Un + 1 
Faciendo u n+1 = quantitati infinite magnae, aequatio in relationem 
a cl. Borchardt inventam transit, scilicet in 
det. j 
| 1 
1 
1 i 
1 (a Y + u) 2 ’ ( a 2 + u f ’ 
(i a n + u ) 2 \ 
det. -1 
( 1 
1 
L i 
\ a x + u ’ a 2 + u ’ 
u=u { , =u 2 , = . 
’ a n + u 1 
. . , Un 
2 
1 
(a 1 + u 1 )(a 2 + u 2 ) . . . {a n + 
Bellavitis (1857, June). 
[Sposizione elementare della teorica dei determinant!. Mem. . . . 
Istituto veneto . . . , vii. pp. 67-144.] 
Bellavitis reaches the subject of the difference-product in § 47 
of his exposition, and his proof of the results dealt with in the 
* Previous suggestions of such a determinant appear in Binet’s paper of 
1837 and Joachimsthal’s of 1854. 
