372 
Proceedings of Royal Society of Edinburgh. 
SESS. 
determinants, of which n{n - l)(?a - 2) would be non-evanescent ; 
and that these could he grouped into sets of six and condensed, 
the ultimate result being 
s 2 1 
"2 
’3 
u 3 ”4 
S, S. 
— | (2g X 2 )(x 3 %\)(x 2 | ( X X\){x — X 2 )(x 
A large generalisation is then made, the exact words being 
“ Genau eben so heweist man folgenden allgemeiuen Satz : 
Bezeichnet man die Potenzsumme x\ + x\+ . . . durch s i ■ 
ferner das Quadrant des Productes, welches aus den \i{i — 1) 
Differenzen der i Grossen x 1 , x 2 , . . . , x { gebildet ist, durch 
8(aJ 1 , x 2 , ... , x { ), so ist 
s o 
s i 
§2 
• • • s a- 1 
1 
*1 
^2 
h 
• • • * a 
X 
S 2 
S 3 
*4 * 
• • • S a+1 
X? 
— ^(*^ 1 ) x 2 r • • • ) x^ix 
-*,) 
{x — x 2 ) . . 
. . ( x-x a ) 
S a+ 1 
5 a+2 • 
• • • S 2a-1 
x a 
WO 
die 
rechte 
_. . . n(n - 1) . . . 
Seite erne Summe von 1 ~ — 
n- a + 1) 
... a 
ahnlich gebildeten Gliederen enthalt.” 
The known result obtained from this by equating coefficients of 
x a is pointed out : also the extension 
s 0 s. 
s 2 ... 
^a-1 
1 
s, s 2 
s 3 ... 
• S a 
X 
- S 2 s 3 
s 4 ... 
• S a+1 
*2 
— €^€ 2 . . . € a . S^Xj^ , X 2 , 
...,x a ) 
(x - x 1 )(x - x 2 ) . . . 
. (x - x a ) y 
Sa S a+1 
®a+2 • • • 
^2a-l 
X a 
where Si 
= ^x\ + 
e 2 x 2 + . 
. . + e n x l n . It may be noted 
that the 
reason for discussing such determinants is that the series of them 
obtained by giving a the values n, n--l , n - 2 , . . . , 2 , 1 , 0 
is put forward (p. 400) as a substitute for Sturm’s series of 
functions.* 
Towards the end of the paper (§ 17, p. 414) the determinant 
* In this connection papers by Cayley (1846) and Borchardt (1845) are 
referred to, but no mention is made of Sylvester’s (1839). 
