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

CONTINUANTS (RAMUS, 1856)

427

RAMUS, C. (1856, March).
[Determinanternes Anvendelse til at bestemme hoven for de
convergerende Brijker. Oversigt... danslke Videnslk. Selsk.
Forhandl.... (Kjobenhavn), pp. 106-119.]
Ramus' introduction consists in recalling the result of the
application of determinants to the solution of a set of linear
equations, his mode of stating the result being that given by
Jacobi in the De forrnatione... of the year 1841,-that is to
say, he takes for his set of equations
co yo + aj0y,1 - c+ 0a,2 +.. ~,,O =e UT
ao yo + a,liy, + aC21y2 ~.. ~ a1,,,, 1 't
aOtZyO + Cal~y,1 + Ca2 nY2 ~.+. + aCY = 'j,
and puts the solution in the form
Rn?/,= A,Ouo + Al-lut ~ Arkt2 + ~ Aq2tn, (w)
where
R71 = E —aOaj1a... a",
A'= E — a~c0Oct11.. a'- ctIa'+'.. aK~K~*.
a1 ~ ~ ~
Ai- C~ CC 0 1 i-1 Ki+1 - 1+
A'=... (tji ajaCL a
K E 0 1cl.. i- i- K-1l Kfl1 1
He then recalls the further fact that if yol,?/i,?~ ~~~,Y be the
numerators of the convergents of the continued fraction
a~o + i b2b7
a, + a29 + + aC,,
"tIt is in this mode of writing A~, namely, with the negative sign, that Jacobi's
peculiarity consists. Not content with removing from R,, the row and column
in which a occurs and prefixing to the minor thus obtained the sign-factor
(-1)i+l+, he takes the further step of moving the row with the index K over
K - i + 1 rows, thus arriving at
i-i i+1 K-i K K+1 n
A`= - a~a.. a a.a a a aS
Of course there is at this second step the option of moving the column with the
index i over K - i + 1 columns, and this Ranus does.