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

ALTERNANTS (JOACHIMSTHAL, 1856)

179

where U cannot contain any of the differences of the u's, and in
its order-number cannot exceed n(n + 1)- n, i.e. n2, Joachimsthal
concludes that V1 and U can only differ by a factor dependent on
the a's. He thus has the two results
J* AA2.. A~ = A(u1, 2...,?n+1)* V1
and V1, = U =. A1A2... A1, 2, 2..., i+ 1]
where ~ is a rational function of the a's: and by combining the
two there is deduced
J = +[1, 2,A, ]. 1.., 2 n+)
A,,... A
At this stage, we are told, the investigation rested for five years
until the publication, in 1855, of Borchardt's paper in the Berlin
Monatsbericht. Taking a hint from this, Joachimsthal, in order
to determine g, multiplied both sides of this result by the product
of all the denominators occurring in the diagonal of J, and then
put u, = -ac, U2 = - 2,.., t=,= -an. The left-hand side was
thus changed into
1 0.... 0 0
0 1.... 0 0
0 0.... 1 0
1 1 1
(aC + ^f+1) (a+2 + ' 'm)2 (,+,SI + l+ )21
or 1; the second factor of the right-hand side, being equal to
(Ul - n+1(2 ) * * ('B - n+l) ' a(l, I,2,.., J,),
was changed into
(- a ) l + x l) +i)... (2 +l)' (- ) i(\-,l (t, (,...+I, a);
and the third factor [1, 2,...., n +l1]/A1A2... A, into a fraction
with the numerator 1 and with the denominator
(aCl-2- ) (C1 -a 3)... (a, -an) (C + Un++i)
~ (a2 -a)(ct -a3)... (a2 - n) (C2 + z,+1)
~ (0a3-a 1)(a3 - 0).. (a3 - n,) (3 + n,,+l)
(aC,-c,) (a,, - c).. * (,,-, - )(a,, +,+,1)