556 
Proceedings of Royal Society of Edinburgh. [sess. 
Rothe (1800). 
[Ueber Permutationen, in Beziehung auf die Stellen ihrer 
Elemente. Anwendnng der daraus abgeleiteten Satzen auf 
das Eliminations - problem. Sammlung combinatorisch - 
analytischer Abhandlungen , herausg. v. C. F. Hindenbuvg ) 
ii. pp. 263-305.] 
The position of Rothe was quite different from that of Lagrange 
and Gauss, as his paper dealt explicitly with determinants (or, 
rather, with the functions afterwards known as determinants), and 
the case of axisymmetry is definitely referred to, although not by 
name. 
His one theorem may be illustrated by the case where the 
number of given equations is 4, and is then to the effect that if 
we have 
where the array of coefficients on the left is axisymmetric, then 
the same peculiarity of axisymmetry must make its appearance in 
the derived set which gives each of the %’s in terms of the four s’ s. 
Starting with the more general set of n equations 
n\’x x + n2’% 2 + • • • + nn m x n = sj , 
and denoting the determinant formed from the coefficients on the 
left by N, and the cofactor in N of any coefficient <pg by f 'pq, he 
proves in Laplace’s method that 
fll*^ + f21’s 2 + • • • • + {n\’s n = N’ajj' 
fl n'Sj + f2 n’s 2 + •••• + f nn m s„ = N'x n , 
where, be it observed, the coefficients of Sj are not the cofactors of 
the coefficients % 1 in the original set of equations but the cofactors 
ax 1 + bx 2 + cx 3 + dx± = s 1 
bx Y + cx 2 + fx 3 + gx^ = s 2 
cx 1 + fx 2 + h %2 + ix 4 = s 3 
dx 1 + gx 2 + ix 3 + jx 4 = sj 
ll’^j + 12-aj 2 + • • • + 1 n'x n - 
21’aq + 22-^. 2 + - • • + 2 n'x n = s 2 
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