374 Proceedings of Royal Society of Edinburgh. [sess. 
where A is the determinant-form of the difference-product of 
x 1 , x 2 , . . . , x n . No explanation of the statement is given, nor 
the mode of arriving at it. All is made clear, however, if we 
note first that by x on the right hand is meant any x chosen at 
will from the set x Y , x 2 , . . . , x n : second, that the differential- 
quotient on the left is intended to stand for the cofactor of the 
(n - l) th power of that particular a: in A, and therefore merely 
denotes the difference-product of a certain n - 1 of the cc’s. 
What the statement thus gives us is an alternative form for the 
square of the difference-product of n - 1 quantities. 
If we use column-by-column multiplication, and put s r for 
a r 4- /3 r -1- y r + 8 r , we clearly have 
4 s 4 s 2 
5 $2 Sg 
s 2 *, s 4 
1/8^1 
1 
P 
P 2 
1 
a 
a 2 
1 
P 
P 
1 
1 
1 
o 
T 
1 
8 
8 2 
1 
a 
a 2 
2 
1 
7 
O 
y 
1 
8 
8 2 
, 
and so the result is established. It will be observed that the 
chosen letter which occurs most conspicuously in the new form 
thus obtained for £(a, y, 8) is one which the expression is quite 
independent of. Further, by performing on this new form the 
operations 
col 4 — col 4 , col 2 — /3 col 4 , col 3 — /3 2 col 4 , 
we return to the more natural form 
£( a > y, S) 
3 a + y + S a 2 + y 2 + S 2 
a + y +8 a 2 + y 2 + S 2 a 3 + y 3 + 8 3 
1 a 2 + y 2 + S 2 a 3 + y 3 + 8 3 a 4 + y 4 + 8 4 
Borchardt (1855, March). 
[Bestimmung der symmetrischen Yerbindungen vermittelst 
ihrer erzeugenden Funktion. . Monatsber . . . . Akad. d . 
Wiss. zu Berlin , 1855, pp. 165-171 : Crelle’s Journ ., liii. 
pp. 193-198 : Gesammelte Werke , pp. 97-105.] 
The generating function in question is 
