Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/274

XIX.

FOURIER'S SERIES.

[Nature, vol. lix, p. 200, Dec. 29, 1898.]

I should like to add a few words concerning the subject of Prof. Michelson's letter in Nature of October 6. In the only reply which I have seen (Nature, October 13), the point of view of Prof. Michelson is hardly considered.

Let us write $f_{n}(x)$ for the sum of the first $n$ terms of the series

$\sin x-{\tfrac {1}{2}}\sin {2x}+{\tfrac {1}{3}}\sin {3x}-{\tfrac {1}{4}}\sin {4x}+{\text{etc.}}$

I suppose that there is no question concerning the form of the curve defined by any equation of the form

$y=2f_{n}(x).$

Let us call such a curve ${\text{C}}_{n}.$ As $n$ increases without limit, the curve approaches a limiting form, which may be thus described. Let a point move from the origin in a straight line at an angle of 45° with the axis of X to the point ($\pi ,\pi$), thence vertically in a straight line to the point ($\pi ,-\pi$), thence obliquely in a straight line to the point ($3\pi ,\pi$), etc The broken line thus described (continued indefinitely forwards and backwards) is the limiting form of the curve as the number of terms increases indefinitely. That is, if any small distance $d$ be first specified, a number $n'$ may be then specified, such that for every value of $n$ greater than $n',$ the distance of any point in ${\text{C}}_{n}$ from the broken line, and of any point in the broken line from ${\text{C}}_{n}$, will be less than the specified distance $d$.

But this limiting line is not the same as that expressed by the equation

$y={\text{limit}}_{n=\infty }2f_{n}(x).$

The vertical portions of the broken line described above are wanting in the locus expressed by this equation, except the points in which they intersect the axis of X. The process indicated in the last equation is virtually to consider the intersections of On with fixed vertical transversals, and seek the limiting positions when $n$ is increased without limit. It is not surprising that this process does not give the vertical portions of the limiting curve. If we should consider the intersections of ${\text{C}}_{n}$ with horizontal transversals, and