411 
RELATIONS OF TWO ALGEBRAICAL FUNCTIONS. 
Section III. is devoted to a determination of the values of the preceding formulae 
for the residues and multipliers in the case applicable to M. Sturm’s theorem, where 
^{x) becomes the differential derivative of/x. It becomes of importance to express 
the formulae for this case in terms of their roots and factors of fx alone, without 
the use of the roots and factors oi fx, which will of course be functions of the 
former. 
By selecting a proper form out of the polymorphic scale, the fractional terms of the 
series for each residue in this case become separately integral, and we obtain my well- 
known formulae for the simplified residues (Sturm’s reduced auxiliary functions) in 
teinrs of the factors and the squared differences of partial groups of roots. This is shown 
in art. (35.). In art. (36.) the multiplier offx in the syzygetic equation is expressed 
by formulae of equal simplicity, and in a certain sense complementary to the former. 
This method, however, does not apply to obtaining expressions for the multiplier of 
/r in the same equation in terms of the roots and factors of/r; for the separate 
fractions whose sum represents any one of these factors it will be found do not 
admit of being expressed as integral functions of the roots and factors. To obviate 
this difficulty I look to the syzygetic equation itself, which contains five quantities, 
viz. the given function, its first differential derivative, the residue of a given degree, 
and the two multipliers, all of which, except the multiplier of fx, are known, 
or have been previously determined as rational integral functions of the roots and 
factors oi fx. I use this equation itself for determining the fifth quantity, the multi- 
plier in question. To perform the general operations by a direct method required 
for this would be impossible ; the difficulty is got over by finding, by means of the 
syzygetic equation, the particular form that the result must assume when certain 
relations of equality spring up between the roots of fx-, and then, by aid of these 
paiticular determinations, the general form is demonstratively inferred. 
This investigation extends over arts. (38.), (39.), (40.), (41.), (42.), (43.). It turns 
out that the expressions for the multipliers 6f/r are of much greater complexity than 
for the multipliers fx or for the residues. Any such multiplier consists of a sum of 
parts, each of which, as in the case of the residues and of the factors of fx, is affected 
with a factor consisting of the squared differences of a group of roots ; but the other 
factor, instead of being simply (as for the residues and factors before mentioned) a 
product of certain factors oi fx, consists of the sum of a series of products of sums 
of powers by products of combinations of factors of fx, each of which series is 
affected with the curious anomaly of its last term, becoming augmented in a certain 
numerical ratio beyond what it should be, in order to be conformable to the regular 
flow of the preceding terms in the series*. 
I he fourth section opens with the establishment of two propositions concerning 
The syzygetic multipliers are identical with the numerators and denominators (expressed in their simplest 
form) of the successive convergents to the continued fraction which expresses 
fx 
