362 Proceedings of Royal Society of Edinburgh. 
[sESS. 
X 1 + X 2 + . . t . + Xn — 1 i 
«1 — 2/l «2“2/l ' a n-Vi 
X l + X 2 + + Xfl _ X ' 
a 2~y2 ’ a n-y 2 
which Binefc had, explicitly dealt with four years before. 
Borchardt (1845, Jan.). 
[bTeue Eigenschaft der Gleichung, mit deren Hiilfe man die 
secularen Stor ungen der Planeten bestimmt. Crelle’s Journ ., 
xxx. pp. 38-45 : Gesammelte Werke , pp. 3-13.] 
For the present this paper is onty noteworthy as containing the 
square of the difference-product in the form of a determinant of 
the particular type soon after to be named <s persymmetric ” by 
Sylvester. M being used to stand for 
{2 
Borchardt says “Es wild also M die Determinante aus dem 
System 
. s o 
. 
s 2 .... 
S n - 1 
. S 1 
s 2 
% . . . . 
s 2 
s 3 
*4 .... 
S n+1 
S Jl-l 
S n 
6 n+l • • • • 
S 2n-2 
+ . 
; +9 m .” 
Rosenhain (1845, Sept.). 
[Neue Darstellung der Resultante der Elimination von z aus 
zwei algebraische Gleichungen f(z) = 0 und cf>(z) — 0, . . 
Crelle’s: Journ., xxx. pp. 157-165.] 
Although the subject of alternating functions is incidentally 
dealt with in Rosenliain’s paper (p. 161), nothing of importance 
occurs. The identity * 
* A still better form for tlie right-hand number is 
r=m + 1 
i • • • > ®s) . j ^m+2 > . • • , U n ) 
s=l , 2, .... m 
