268 Proceedings of the Boyal Society of Edinburgh. [Sess. 
and the second is 
ajalal + plpifil + ylytyt = aipiyi + a\ply\ + a§/%| , 
the latter’s existence being due to the fact that the right-hand member of 
the former is not altered by the interchange 
a 3 
7i iJ * 
Borchardt, C. W. (1845, January). 
[Neue Eigenschaft der Gleichung, mit deren Hiilfe man die secularen 
Storungen der Planeten bestimmt. Crelles Journ., xxx. pp. 38-45 : 
or, in an extended form, Journ. ( de Liouville) de Math., xii. 
pp. 50-67 : or Werke, pp. 3-13.] 
Borchardt’s “ new property ” is the naturally desirable generalisation of 
Rummer’s identity. In his mode of designating the equation* he does not 
follow Rummer and Jacobi, but goes back to Cauchy (1829), the implied 
reference being to Laplace’s Mecanique Celeste, partie i., livre ii., § 56 
(1799). 
The set of equations, from which by elimination there is obtained the 
equation referred to in the title, being 
9*i = 
+ a 12 x 2 + . 
• • + %A 
~ 
^21*^1 ^22^2 * 
. + a 2n x n 
^ • 
£ 
ii 
a nX x 1 + a n2 x 2 + . . 
. + CL nn X n ^ 
where a ik = a u , Borchardt multiplies the two sides of each equation of the 
set by g, and then on the right-hand side substitutes for gx v gx 2 , . . . , gx n 
their equivalents as given. There thus results the new set 
= a ( ^x x + a[ f a? 2 + . . . + " 
fa 2 = + ofe + • • • + d$lx n 
g 2 x n = afc + a$x 2 + . . . + a ( nlx„ J 
S=?l 
whereat = w = Repeating this operation he finds gener- 
S=1 
ally that 
* The most appropriate designation would seem to be “Lagrange’s determinantal equa- 
tion,” because of this mathematician’s early (1773) and successful investigation of the cubic. 
See his CEuvres completes , iii. pp. 600-603. 
