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
seek the limits which they approach when n is increased indefinitely we should obtain the vertical portions of the limiting curve as well as the oblique portions.
It should be observed that if we take the equation
$y=2f_{n}(x),$


and proceed to the limit for
$n=\infty ,$ we do not necessarily get
$y=0$ for
$x=\pi .$ We may get that ratio by first setting
$x=\pi ,$ and then passing to the limit. We may also get
$y=1,x=\pi ,$ by first setting
$y=1,$ and then passing to the limit. Now the limit represented by the equation of the broken line described above is not a special or partial limit relating solely to some special method of passing to the limit, but it is the complete limit embracing all sets of values of
$x$ and
$y$ which can be obtained by any process of passing to the limit.
New Haven, Conn., November 29 [1898].
[Nature, vol. lix, p. 606, April 27, 1899.]
I should like to correct a careless error which I made (Nature, December 29, 1898) in describing the limiting form of the family of curves represented by the equation
$y=2\left(\sin x{\frac {1}{2}}\sin 2x...\pm {\frac {1}{n}}\sin nx\right)$

(1)

as a zigzag line consisting of alternate inclined and vertical portions. The inclined portions were correctly given, but the vertical portions, which are bisected by the axis of
${\text{X,}}$ extend beyond the points where they meet the inclined portions, their total lengths being expressed by four times the definite integral
$\int _{0}^{\pi }{\frac {\sin u}{u}}du.$


If we call this combination of inclined and vertical lines
${\text{C,}}$ and the graph of equation (1)
${\text{C}}_{n},$ and if any finite distance
$d$ be specified, and we take for
$n$ any number greater than
$100\div d^{2},$ the distance of every point in
${\text{C}}_{n}$ from
${\text{C}}$ is less than
$d,$ and the distance of every point in
${\text{C}}$ from
${\text{C}}_{n}$ is also less than
$d.$ We may therefore call
${\text{C}}$ the limit (or limiting form) of the sequence of curves of which
${\text{C}}_{n}$ is the general designation.
But this limiting form of the graphs of the functions expressed by the sum (1) is different from the graph of the function expressed by the limit of that sum. In the latter the vertical portions are wanting, except their middle points.
I think this distinction important, for (with exception of what relates to my unfortunate blunder described above) whatever differences of opinion have been expressed on this subject seem due, for the most part, to the fact that some writers have had in mind the limit of the graphs, and others the graph of the limit of the sum. A misunderstanding on this point is a natural consequence of the usage which allows us to omit the word limit in certain connections, as when we speak of the sum of an infinite series. In terms thus abbreviated, either of the things which I have sought to distinguish may be called the graph of the sum of the infinite series.
New Haven, April 12 [1899].