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

LESS COMMON SPECIAL FORMS (JOACHIMSTHAL, 1856) 447

JOACHIMSTHAL, F. (1856, September).
[De aquationibus quarti et sexti gradus qua in theoria linearum
et superficierum secundi gradus occurrunt. Crelle's Journ.,
liii. pp. 149-172.]
Joachimsthal, requiring the use of the so-called "Binetf's"
identities, devotes section iii. of his paper to them, combining
them in one proposition, and showing more or less satisfactorily,
after the manner of Meier Hirsch, how the proof of each case
can be made dependent on the previous case. His proposition
is-There being m rows each of z quantities
U 1 a2...
P1 2t ~ ~~ P
01 i3.... /Xz
AI l2.... 1-4z
and z being not less than m, the sum
la1/2...AmI
consisting -of z (z - 1) (z - 2)... (z - m +1) terms, can be expressed
as an integral function of the sums arranged in the following
rows:
Lla, a la~y......... EX~u
7,ajoly, lal,3,6,............... ftchlml
~Ga1I1Y1~ ~~a1/38
each smm consisting of z terms: further, the said function when
z <m vanishes identically.
To prove the proposition when m =3 he takes the previous
case
Z(11132 = -a 10 i-3l,81 (x2)
and multiplies both sides by Eyl, thus obtaining
Ia1A2Y3 + 7a1Y1/32 ~ Y,31ya2 = Ea1/1P,,Y1 - Ya1/31. ~Yil