52 Proceedings of the Royal Society of Edinburgh. [Sess. 
Freuchen, P. (1863). 
[To Determinanter af nte Grad. Math. Tiotsskrift, v. p. 42.] 
The two determinants are the cases of Ferrers’ of the year 1855, in 
which a 1 = a 2 = a 3 — . . . . 
Roberts, M. (1864, March). 
[Question 694. Nouv. Annates de Math. (2), iii. pp. 139-140. Solutions 
by L. Ferrara, G. Torelli, in Giornale di Mat., ii. pp. 95-96, p. 191 : 
solutions by A. Smet-Jamar, M. Cornu, H. Picquet, in Nouv. 
Annates de Math. (2), iii. pp. 395-399, and by “ Mirza-Nizam ” in 
(2), iv. pp. 500-504.] 
Roberts’ result is essentially the same as Ferrers’ second, but is expressed 
more suggestively, namely, 
aj -1 -1 .... -1 
-1 a 2 -1 .... -1 
- 1 - 1 03 .... - 1 
/(1)-/(1) 5 
where f(x) = (x + a 1 ) . . . (x + a n ). 
Ferrara and Smet-Jamar develop the determinant in Cayley’s manner 
1847, obtaining /(l)— /'(l) in the form 
(1-w) + (2-rc)2 a i + (3-71)^^^ + .... + a x a 2 . . .a n ; 
Cornu and Picquet subtract the last column from each of the others and 
obtain the development in the form 
(l+o 1 )(l+a 2 )...(l+a w ).ri— + ■ + ... 
L \ 1 + Oi 1 + a 2 i + J 
Torelli, after generalising the determinant by putting — x everywhere for 
— 1, writes it in the form 
l) x 
8 
1 
O 
8 
1 
o 
1 
o 
8 
1 
<M 
a 
+ 
8 
0 - x 
0 
1 
8 
8 
1 
O 
(x + a s ) - X 
which, if expressed as a sum of determinants with monomial elements, gives 
(x + a x )(x + a 2 ) . . . (x + On) - x'^ (x + a 2 )(x + a 3 ) . . . (x + a n ) 
i.e. f(x) - xf(x) : 
and “ Mirza Nizam ” obtains this wider result of Torelli’s by following 
Cornu and Picquet. 
