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.

Newton's rule for expanding at an arbitrary point.

385

in which we can retain for the root )ql some one of its q1 values,
while zq assumes all its q1 different values.
Accordingly, when all the roots of the equations for A are simple,
the /l, / 2 - PI... h - Pi-1 expansions which belong to the i sides of
the polygon resolve respectively into /k cycles each having q1 values,
into k2 cycles each having q2 values,..., into ki cycles each having
qi values; so that in fact all these expansions are necessary in order
to exhibit the h values of w that vanish for z - 0.
Relatively to each simple root I, the next term in the expansion:
P p+1l P 1
_ -~ 2;v/ St -- ' \'
w = —vz v * * *.. vz~ + 1 z2 + *q-~
is obtained by attending to the term of the next dimension of S in
substituting this series for w. The coefficient of this term equated
to zero, presents a linear equation for determining v1.
To investigate the signification of a multiple root of the equation:
Al,0+- Z+Aa,fk +- A,,k, 0,
we suppose it to have j roots equal to A; then each of the forms:
1 PI
W == l.
must be initial term in qlj expansions; but these again will resolve
into certain cycles. In order to perceive that they do, let us substitute
Z q') w == ( iq + w'Jz
in the original algebraic equation. This is thereby converted into
an equation between z' and w', which, developed by powers of these
quantities, must have j roots w' vanishing along with ' -= 0. In fact
{ A1,oZ1+ Af zue -t- Aafzlw'W } + { SA,1,ze'wO'+ A0,^w1 +1-F (Z,w! },
in which the first brackets contain all the terms of lowest dimensions,
passes over, as regards these terms, into:
Al oz'l + A ge 'q + w ') 'iP- + Afl,z'aq,(IQl + tVw), glpj
or, dividing by the lowest power of z', the value of whose exponent is
li - t ql + fiP1 - 1 ql + fipPi, into:
( 1 ) (1 )/~1
A1,0 + 2Aa,/,(qt + )Y+ A+a,( QI + Z),
and of these, the term independent of w' vanishes, because X is a root
of the equation:
_ 1
A1 A, + AA,,q -- = + AAa,,Aq + -Aa, A,,lk -. 0.
HARNACK, Calculus. 25