The theory of determinants in the historical order of development, by Sir Thomas Muir.

AXISYMMETRIC DETERMINANTS (BRIOSCIH, 1855) 141
column is a sum of multiples of the other columns, he multiplies
it by - of itself, namely by
1 I(x6-x )X O' 1 xi X, zi
1 7(X,6 2)2 X2 2 X2 112 Z2
1 (X,-X)2 X5 y5 g
1 1
and, putting d61,... for y (x6 -x)2,..., obtains the relation
d6l2 d6ld62+dl2. d6ld65 -4 15 1 + 1
d6(c162+ d12 d622.. c.62d65 +125 d62 +
d61d65+d,15 d62d65 + d25... d652 d65 + 1
c161+1 d62+ 1... d65+ 1 1
which degenerates into Cayley's result when we put
X6' Y1' Z6 = r1, y1, z1,
and make certain easy transformations.
In a similar manner the relation between the distances of
five points on an ellipsoid is found, and the relation "entre les
plus courtes distances respectives et les inclinaisons mutuelles de
sept lignes quelconques."
The second paper contains nothing new.
FERRERS, N. M. (1855, Dee.).
[Two elementary theorems in determinants. Quarterly Journv.
of Math., i. p. -364; or Nouv. Arnvales de Math., xvi. pp.
402-403, xvii. pp. 190-191.1
The first theorem referred to is
1 1 1-c~... 1
1 1+a1 1.. 1
1 1 1+2.. 1 a1a2
1 1 1... a