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.

~ 202. 203. Newton's rule for expanding at an arbitrary point.

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simple. The process would only fail to lead to the desired end, it'
after a certain stage the multiplicity of the root remained always
equal to the degree of the equation in question. But this supposition
contradicts the hypothesis that the form considered is irreducible.
For, assuming, as we may without any restriction of generality,
that from the outset the equation:
A.,o + Z'Aa,,Ak + Aa,,a,. 0,, O'
has k1 roots equal, and is therefore:
Ac,,f, (A - 0)k_ =- 0
the form of the original equation must be
" A a,,,, (wvq, - A0o'P,) + p (, ( w) (0.
By substituting: - ==,'q, tw =to= +"' w')z'P, and dividing by the
factor 'q(,+i), )-= 'q,, we obtain the equation between ' and zv':
Aacil K + 0) -- A + (Z', Wv') = ),
where t(z', v') denotes the terms proceeding from (p; the lowest power
of w' is wv'k,. By hypothesis this equation is to give only a single
root, to be counted k, times, for the expansion of w'. But this requires
that the corresponding polygon should reduce to a single right line
and that therefore the corresponding equation should be:
A(' - Az'p')k, + q, 1(, ) o0.
Now supposing the further substitutions carried out, and that we
found every time an equation of degree k7 between w and z, there
should exist an expansion:
L p,,L- P+lP'+P"
Wv =;ql — S 'z q +;" q" + -...,
that must be valid however great the exponents of z become. By this
expansion 7k roots of the equation would be expressed. The algebraic
form would therefore have k1 equal roots within a certain domain of
z and consequently in general. Since, starting from a value w, we
can establish an irreducible form (~ 199) whose root is w, it follows
that the original algebraic form on our hypothesis must have contained
as a factor the kth power of an irreducible factor or must have been
equal to the 7th power of such a form.
203. The example in the foot-note to ~ 202, p. 381:
A9,oz9 + A7,1z7 vq + A5,2 Z5 -j2 + A4,4 z4w4 +A1,z 5 +Ao,Aw7 W+ _ (z, w) - 0,
where
9(z, w) A= Al,Gzwo + A7,2z7w2 -- A5,4z5v4 + Ao,8sS +l A10,010,
in which seven values of wv vanish for z = 0, leads, as the polygon
fig. 20 shows, to the three aggregates of terms of equal dimensions:
25*