398 Proceedings of Royal Society of Edinburgh. [sess. 
leaving as cofactor a determinant of the ( n - l) th order whose 
diagonal elements are all 1 -n and non-diagonal elements all 1. 
On this new determinant he then performs the operations 
rowj + row 2 + . . . + row M _ 1 , 
row 2 - 4 - rowj , row 3 + rowj , 
and so finds its value to he 
(-If- 1 • n n ~ 2 
which gives for the circulant with which he started the value 
( - • ( nd) n ~ l • (a + d\ . 
Baehr (1860). 
[Solution de la question 432. Nouv. Annales de Math., xix. 
pp. 170-174.] 
After dealing as we have seen with the circulant whose elements 
are in equidifferent progression Baehr proceeds to the circulant 
whose elements are in equirational progression, namely 
C(a , ar , ar 2 , . . . , ar n ~ x ) . 
This he first changes into 
a n • C'(l , r, r 2 , . . . , r n ~ x ) 
and then into 
( - l)**- 1 * • a n • C (r n ~ x , r n ~ 2 , . . . , r , 1) . 
On the determinant thus reached the operations 
r row x - row 2 , r row 2 - row 3 , . . . . 
are performed, with the result that its value is found to he 
( - If" 1 • (1 - py - 1 , 
and thence the value of the original circulant to he 
^ 1) . d n (j- n _ If -1 . 
It is worth noting that instead of the last set of operations we 
might substitute with advantage the set 
row„ - r row n _! , row^ - r row„_ 2 , . . . . ; 
also, that Baehr’s circulant is a special case of that referred to 
under Cremona’s third lemma. 
(. Issued separately November 16 , 1906 .) 
