392 Proceedings of Royal Society of Edinburgh. [sess. 
On d4duit, de ces diverses formules 
A = (w — 2)( - 2) n_1 .” 
This result, which at a later date would have been written 
C(-l, 1, 1, 1) = (»-2)(-2)»- 1 , 
and which, we may point out in passing, could also he reached by 
the operations 
row 1 + row 2 + . . . +row n , 
remove factor to - 2 , 
row n - row n _! , row n _ x - row n _ 2 , .... 
is then attempted to he generalised (§18) by withdrawing the 
restriction as to the number of negative units in a row. The 
reasoning, however, seems to have been incautiously conducted, 
the extension arrived at being 
C( - 1 , -1, 1, l)„- r = (»-2p)(-2)"- 1 , 
where the number of consecutive negative units in the first row 
is p, and the number of positive units n-p. 
Catalan then passes (§19) to the consideration of the similar 
circulant whose first row consists of p consecutive positive units 
followed by to— p zeros, separating the investigation into two 
parts, (1) the case where p and to have a common factor other 
than unity, (2) where they are mutually prime. In the former 
case he shows that the equations which have the circulant in 
question for determinant are “ ind6terminees ou incompatibles ” : 
in the latter case he shows that the equations are determinate. 
He thereupon goes on to supplement the information in the 
second case by proving that the circulant is equal to p : he omits, 
however, any similar proof that in the first case the circulant is 
zero. 
Lastly, he attacks the general circulant, or, as he calls it, “le 
determinant du systeme 
CL-^ CLcy (f/g • • • 
a 2 a 3 a 4 ... 
«3 «5 ‘ ‘ ' a 2 
a n a 1 a 2 ... 
