312 Proceedings of the Royal Society of Edinburgh. [Sess. 
Borchardt, C. W. (1855, 5/8). 
[Bestimmung der symmetrischen Verbindungen vermittelst ihrer 
erzeugenden Function. Monatsb. . . . Akad. d. TPiss. zu Berlin, 
1855, pp. 165-171 : or Grelle’s Journ., liii. pp. 193-198 : or 
Gesammelte Werke, pp. 97-105.] 
Having already fully dealt with this paper under the heading 
Alternants, it suffices merely to recall the identity therein given, namely, 
yf - 1 . J_. 1 ) x y ( " ± 1 . 1 ... M 
= Z( ± (j_ a)2 • ( i^)2 • • • m) > 
where the first factor on the left differs from the determinant which is its 
cofactor merely in having the signs of all its terms positive. 
Joachimsthal, F. (1856, September). 
[De sequationibus quarti et sexti gradus quse in theoria linearum et 
superficierum secundi gradus occurrunt. Grelle’s Journ., liii. 
pp. 149-172.] 
Joachimsthal, requiring the use of the so-called “ Binet’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 
“l 
a 2 . . . 
• 
p 1 
P 2 
■ ft 
A 2 . . . 
• A, 
Hi 
H • • • 
• Hz 
and z being not less than m, the sum 
^^ a l/? 2 • • -Mm 
consisting of z(z — l)(z — 2) . . . (z — m + 1) terms can be expressed as an 
integral function of the sums 
2 a i , , . . . . , Syttj 
Wi » 2a i7i > • ■ • ■ » 
2a i/hri » > • • • • j 2 *i*i/*i 
2a ifryi • • • /*i » 
