Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/275

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.

J. Willard Gibbs.

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.