458 MR. SYLVESTER ON FORMULA RELATING TO STURMS THEOREM. 
since (— which gives the well-known formulae (enunciated by me in the 
London and Edinburgh Philosophical Magazine for 1839) for expressing M. Sturm’s 
auxiliary functions in terras of the roots of the primitive, and which I therein stated 
were immediately deducible from the general formute (also enunciated m the same 
paper) applicable to any two functions. These more general formute appear to have 
completely escaped the notice of M. Sturm and others, who have used the special 
formute applicable to the case of one function becoming the first differential den- 
vative of the other. , , j- 
Art. (36.). In precisely the same manner, if we form as usual the ordinary syzygetic 
we may find the different values of t given by the complementary formute; and 
using t, to denote the multiplier of the degree i in .r, i. e. appertaining to the residue 
of the degree in x, we have 
^2 • • *^ 171—1 
{x-h,){x-hj...{x-h^;) 
Art. (37.). Thus, if we make 
It is evident from the form of/'.^ that it possesses relative to /r, the same pro- 
perty as fx, I mean the property that when a; is indefinitely near to a real root of 
fx, and is passing from the inferior to the superior side of such root, like wdl 
pass from being negative to being positive, or in other words, /> and /'.r have 
always the same sign in the immediate vicinity to a real root of /r. Hence it fol- 
lows that f[{x) might be used instead oifx, to produce, by the Sturiman process of 
common measure, a series of auxiliary functions, which with /a: and / ..r would form 
a rhizoristic series, L e. a series for determining (as in the manner of U. Sturm’s 
ordinary auxiliaries) the number of real roots of Jx comprised within given limits. 
The rhizoristic series generated by this process will, it is easily seen, be (to a con- 
stant factor pr^s) the denominators (reckoning -f I as the denominator in the zero 
place) of the successive convergents to-^ thrown under the form of a continued frac- 
_i_ _i —L- 1 . M. Sturm’s own rhizoristic series, on the contrary (will 
gi — §2 Qn-l ^ 
be to a constant factor prh), the denominators of the convei gents to tie .nveise 
fraction^, which will beof the form K-A h accordingly these two 
