411 
1909-10.] The Theory of Persymmetric Determinants. 
For ourselves we may add that the theorem becomes still more inter- 
esting when it is pointed out that, by reason of the identity 
y\v\ • 
yl-Jn I | y\y \ . . . yl-lyl 
Y, 
.+ Y * 
. + 
where (p(y) — {y — y^){y — y 2 ) . . . (y — y n )> the R’s like the u>’s are all 
expressible as determinants of the order n + m+ 1 , that the determinants 
in both cases belong to the special type known as alternants, and that 
R p differs from w p in the last column only ; — in fact, that 
II 
pf 
1 
X 1 
x[ . . 
^.n+m-1 
xfu.Kx^x) 
+{» 
1 
X^ 
A 
ry.n+m-1 
Mj/(a; 2 - *) 
1 
x 3 
A . . 
^,n+m — 1 
x 3 
X%U 3 l(x 3 - x) 
w p = 
1 
x 1 
xl . . 
g.n+m - 1 
x(ufx 1 - x) 
+i* 
1 
X 2 
x\ . . 
^n+w-1 
xfu.fx^ - x) 
1 
x 3 
x\ . • 
~n+m - 1 
•*3 
x%u 3 {x 3 - X) 
where is the difference-product of x 1 , x 2 
'n+m+V 
Borchardt, C. W. (1847, February). 
[Developpements sur lequation a l’aide de laquelle on determine les 
inegalites seculaires du mouvement des planetes. Journ. {de 
Liduville) de Math., xii. pp. 50-67 : Gesammelte Werke, pp. 15-30.] 
The new section of this paper, which is an extension of Borchardt’s of 
1845 (January), is the third (pp. 54-60), and explains at length how, for the 
purpose of ascertaining the total number of real roots of the equation of 
the n fch degree f(x) = 0, the coefficients of highest powers in the series of 
Sturm’s functions f(x ) , f ± (x ) , f 2 (x ) , . . . may be replaced, according to 
Sylvester, by 
1 J n i 2, { x 2 ~ x lf ’ 2,(^2 “ x l) 2 ( X 3 ~ :^l) 2 (^2 — X \f 5 • • • • 
where x 1 ,x 2 , . . . are the roots, and therefore by 
I s o 
S l > 
s o 
S 1 
S 2 
s l 
h 1 
S 2 
S 3 
^2 
S 3 
S 4 
where s r . = ^ + ^+ • • • +<• All this, however, is practically implied in 
Cayley’s paper of 1846 (August).* 
* The proposition Borchardt is concerned with is of course that The equation f(x)=0 has 
as many pairs of imaginary roots as there are changes of sign in any one of the three series 
mentioned. 
