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.

383

the point x =, y = 0 to the point x = 0, y = h as shall form a
polygon convex to the origin (,t > 0) but concave to all the remaining
points. Hence: turn a right line clockwise round the point x = 1,
y - 0, from being along the axis of abscissae, until it first meets
one or more of the points marked down. Let ca, jj be the most
distant of these from the turning point, then the terms of lowest
dimensions are:
A,0o zl +27Aa,t a w + Aal,, 1 w,
and the corresponding value of p, found from the equation
O=ai +: -El = -1
is a rational number; to indicate that it is not necessarily an integer,
let it be denoted by
pi
where p, and q, are relatively prime. Substituting the value w = vsz,
we find an expression from which the factor z1 can be separated, and
since the term independent of z must vanish of itself, we obtain for the
determination of v the equation of the degree /1:
Al,o +2A,,,V( + A,l^,1vIA = 0.
This presents 0l values for v, finite but not all necessarily distinct;
so that from this first assumption we should obtain the initial terms
of j3 series.
Let us now consider a second side of the polygon, rotating the
right line further from left to right round the point ca,, P, till it meets
one or more of the points marked down. Let the furthest of these
from ac, O,, be c2r, P2. The corresponding terms:
Aba z t -Wi + Aa,eZawa -+ AA,,, V"2a^wA
are then of equal dimensions when we put:
1 - z2 1 92
I +:/ = 2 + 2, or: — e p, _,2 2
P2
and for determining the coefficient v in the expansion w == vs2, on
substituting this value for w, we find the equation of the degree 2 - P:
A,ia, +7Aa,,vi-~iil + Aa2,,2V12-l = 0.
Continuing this process, we ultimately obtain from the last (ith) side
of the polygon, which must pass through the point x = 0, y =- h, the
combination:
A.-li ai-lwi4- +2Aa,:Zawe + Ao,, h, -I _ === ->
i'f h - Yi-1 Ci