An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 201. 202. Newton's rule for expanding at an arbitrary point.

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not be irreducible. Let this term be z, its coefficient A1,o. Let us
then take the term independent of z in which w occurs in the lowest
power. Let it be AohW". Between these two let us arrange all those
terms of the form A,,,a"w whose exponents a and f3 are respectively
lower than 1 and h, while of terms with the same power z (or wfI)
we always take only that one in which the exponent of w (or z) is
lowest. The terms thus selected can then be so arranged that the
series of numbers a shall decrease from 1 to 0, and that the series
of numbers p shall increase from 0 to h. All the rest of the terms,
in case there be any over, may be denoted by (p(, w) so that we
shall write down the aggregate in the form:*)
f(z +, + b) wol +),g + Aa.Zv + l o + Aow + g(, w) = 0
Since the point is by hypothesis k-elementary, a term of the kth
dimension is certain to occur among the bracketed terms. For
z = 0, h values of w vanish, what is required therefore is to obtain
expansions of these h roots in series of ascending powers of z.
Such an expansion begins with the term: wz = vz,..., where t
must be a positive number, integer or fractional. If the series were
known, all the powers of w could be expressed by series valid within
the same circle of convergence, the expansion for wz beginning with
the term: vilfbll. Substituting then these series in the above form, the
expression should vanish identically: i. e. the coefficients of its various
powers of z should be separately zero. But when we substitute
W - vzP' wI- == vflz',
the form of the above expression becomes:
Ai,oez +,Aa + Ao,^,vzh,' + q (a, ve Z);
the substitution of further terms of w only introduces terms whose
dimensions exceed those written down. But even among those written
down, in consequence of our selection, the dimensions of the terms in
Sp(za,vz~) are certainly higher than of those bracketed; for, if there
be a term zawlt in Tp, there is a term inside the brackets in which
at least one exponent is less than the corresponding a or f. If the
value of, were known, those terms within the brackets for which the
exponent is lowest could easily be pointed out; but as, has first to be
found, the inverse questions arise: What assumption as to terms of equal
*) Example: From the equation: A5zzow5 + A5,25 t2 + A 1,6w6 + Ao7w7
+ A7,1z7wo + A4, Z4w4 + Ao8 W+ + AS9,09 + A5,454 + A7,2 7w2 + A10 0 = 0,
in which z = 0, w = 0 is a 6-elementary point, the teris:
A9,0z9 + A7,1z7w + A5,25w2 + A4,4 ZW4 + A1,5ZO5 - + A0,7 w7
have to be singled out; all the rest belong to the aggregate (p(z, w).