MR. SYLVESTER ON FORMULAE RELATING TO STURM’S THEOREM. 
459 
rhizoristic series will be equivalent as regards the number of changes and of combina- 
tions of sign (afforded byeach) corresponding to any given value of x, of which of course 
the c[% are linear functions. This result agrees with what has been demonstrated by 
me by a more general method (in the London and Edinburgh Philosophical Magazine, 
June and July J 853), where it has been proved, by means of a very simple theorem of 
determinants, that the two series 
L _i_ ^ .Jl_ _L_ 1 __L 1 1 
?] ’ ?]- ^ 92 - ”* 9 ]- 92 - % ’"Yn 
and 
1. _L _1_. —L _i_ J_. J- 1 1 2 
([n—l §»— 1 Q.n—2 ^n~l 2 
always contain (for real values of q^, q^, the same number of positive and 
negative signs. 
Art. (38.). Having now determined the general values of and t in the equation 
tf'{x) — Tfx-\-'^=Q as explicit integral functions of the roots of fx, the more difficult 
task remains to assign to r its value similarly expressed. This cannot readily be 
effected by means of substitutions in the general formulae, the method we adopted for 
finding t and ^ ; but all the other quantities except t in the syzyzetic equation being 
integral functions of the roots, it is evident that r also must be an integral function 
of the same, and to obtain it we may use the expression t= — . 
JX 
To obtain the general form of r by direct calculation from this formula would 
however be found to be impracticable ; the mode I adopt therefore to discover the 
general expression for r corresponding to different values of is to ascertain its 
value on the hypothesis of particular relations existing between the roots of fx, and 
then from the particular values of r thus obtained to infer demonstratively its general 
form, as will be seen below. The demonstration of r is unavoidably somewhat long, 
r being in fact represented by a double sum of partial symmetrical functions. 
Using the subscript indices of each function as the syzygetic equation to denote its 
degree in x, we have in general 
./'j: — 0, 
where if we make 
hi X — ^1 h^ X — h.2 X'=-h^, 
h^ — h^—hx—h^, 
we have in effect found 
so that 
and therefore 
and 
we have also /'(x) = (—)’""’. 
