390 
Proceedings of Royal Society of Edinburgh ■ [sess. 
The Theory of Circulants in the Historical Order of 
Development up to 1860. By Thomas Muir, LL.D. 
(MS. received July 9, 1906. Read July 13, 1906.) 
So far as mathematical writers have as yet noted, a set of 
equations of the type 
a l x l + a 2 x 2 + . 
. . + a n x n — u j , 
a n sc 1 4- a Y x 2 + . 
• • +VA = %, 
'>n-l X l + a n X 2 + • 
. . + a n _<pc, n — u s , 
a 2 x x + a z x 2 + . 
. . + a-jX n — u n , 
had not made its appearance in mathematical work prior to the 
year 1846 : and it is almost absolutely certain that before that 
year the determinant of such a set had never been considered. 
It is not at all unlikely, however, that the expression 
a B + b B + c B - 3 abc 
which is the case of the determinant for n equal to 3 had more 
than once turned up in other connections, and that its divisibility 
by a + b + c had been noted : but of this, too, there is no record. 
Catalan (1846). 
[Becherches sur les determinants. Bull, de V Acad, roy 
de Belgique , xiii. pp. 534-555.] 
As has been already explained, about half of Catalan’s paper is 
occupied with an elementary exposition of known properties of 
determinants and with the establishment of a fresh theorem of 
his own, which in his notation might have been written in the 
form 
det. ( A 1 + A 2 4- . . . + A n , A x - A 2 , A 2 - A 3 , . . . , A n _ x - A n ) 
= ( - l) n_1 . n . det. (A 1 , A 2 , . . . , A n ) , 
and which is to the effect that If from a determinant A of the n th 
order we form another A', such that the first row of A' is the sum 
