482 MR. SYLVESTER ON FORMULAE CONNECTED WITH STURM S THEOREM. 
will assume the form 
flfj <^2 
a 
a, 
yl- 
«2 
a, a, 
^2 
2/I+...+ 
Q!j <Z2 • • • ^n— 1 ^nj 
<7l ®nl..2 
if’' 
0-\ 0-2 • • • Oji—\ 
dyCl^ - • 1 
and consequently the number of positive squares in the reduced form of the given 
function will always be the number of continuations or permanencies of sign of the 
series 
1 ; 
ffi 
dl ^2 I 
di ^2 I 
dj d2 ^3 
di 0^2 • • •^Tij 
the several terms of this progression being in fact the determinants of what the 
given function becomes when we obliterate successively all the variables but one, 
then all but that another, then all but these two and a third, until finally, the l^t 
term is the determinant of the given function with all the variables retained. Jhis 
r* I * / ^ TTM TTOrtTol~iloo\ Q TT P 
dj 
dJ 
dJ 
dj 
dx\ 
dxi dx,2 
dx-^ dx^ 
dx-^ dx^ 
dj 
dY 
dJ 
dJ 
dx,^ dx-^ 
dxl 
doC(^ doc^ 
dX:2 ^4 
dj 
dJ 
dj 
dY 
dx^ dxi 
doo^ doCc^^ 
dxl 
dxQ dx^ 
dj 
dj 
dJ 
dY 
d4 dxy 
dx^ dx,^ 
doc^ doc^ 
dxV 
{where all the terms are of course coefficients of the given function expressed as above 
for greater svmmetry of notation), the inertia of / will be measured by the number 
of continuations of sign in the series formed of the successive p-hicipal minor coaial 
determinants (in writing which I shall use in general (r, s) to denote ^J, 
"(1,1) (1,2) (1,3) (1,4) 
1 ,( 1 , 1 ), 
( 1 , 1 ) 
L(2, 1) 
( 1 , 2 )' 
( 2 , 2 ) 
■■I, 1 
2 , 1 
3, 1 
1,2 
2,2 
3,2 
1,3' 
2,3 
3, 3 
( 2 , 1 ) 
(3, 1) 
1(4, 1) 
( 2 , 2 ) 
(3, 2) 
(4,2) 
(2, 3) 
(3, 3) 
(4, 3) 
(2, 4) 
(3, 41 
(4,4). 
and in like manner in general*. 
. I hare giv.a a direct il posteriori demonstration in the London and Edinburgh Philosophical Magarine. 
that the nnLr of continuations of sign in any series formed like the above form a symmetrical matri^ » 
unaffected by any permutations of the lines and columns thereof, which leaves the symmeWy subsisting that 
"to say Jug thi umbral notationl, if 9,,. are disjunctively equal, each to cacli. lu any arbitrary 
1 , 
1 
1 “01 
“0-1 j 
“0, 
“0U “03 1 
«9, ae.2 “03 • • 
1 “01 
1 “01 
“0.11 
«0o “03 1 
“0, “03 “03 •• 
..aei 
is irrespective of the order of the natural numbers 1, 2, 3.... i in the arrangement fi, 
