NA TURE 



[December 29, 1898 



LETTERS TO THE EDITOR. 

 'J'he Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications.'^ 



Fourier's Series. 

 In reply to Mr. Love's remarks in N.VTURE of October 13, I 

 would say that in the series 



y = sin . 



A sin : 



sin (« - i).r + -sin nj-, 



in which - sin «.v is the last term considered, x must be taken 



smaller than TJn in order to find the values n{ y in the immediate 

 vicinity of x = o. 



If it is inadmissible to stop at " any convenient «th term," 

 it is quite as illogical to stop at the equally "convenient" 

 value ir/«. .\i HEKT A. MicHELSox. 



The University of Chicago Ryerson Physical Laboratory, 

 Chicago, December I. 



I SHOfLD like to add a few words concerning the subject of 

 Prof. Michelson's letter in N.^iure of October 6. In the only 

 reply which I have seen (N.^ruRE, October 13), the point of 

 view of Prof. Michelson is hardly considered. 



Let us write /„(.v) for the sum of the first « [terms of the 

 series 



sin .<■ - h sin 2 v + i sin 3.1 — \ sin 4,1 + i:c. 



I suppose that there is no question concerning the form of Ih^ 

 curve defined by any equation of the form 



y = 2/„(.v). 



Let us call such a curve C„. .\s ;; increases without limit, 

 the curve approaches a limiting foim, 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 (ir, ir), thence 

 vertically in a straight line to the point (ir, - tt), thence obliquely 

 in a straight line to the point (3 ir, ir), ts.c. 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 «' may be then specified, such that for 

 every value of n greater than ?;', the distance of any point 

 in C„ from the broken line, and of any point in the broken line 

 from C„, will be less than the specified distance d. 



Pjut this limiting line is not the same as that expressed by the 

 equation 



y = limit 2/ii(.r). 



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 indi- 

 cated in the last equation is virtually to consider the intersections 

 of C„ 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 limit- 

 ing curve. If we should consider the intersections of C„ with 

 horizontal transversals, and seek the limits which they approach 

 when K 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,M, 



and proceed to the limit for n = an , we do not necessarily get 

 y = o (ot X = ir. We may get that ratio by first setting .v = ir, 

 and then passing to the limit. We may also get ^ = 1, .r = ir, 

 by first setting _y = I, 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 .<■ and y which can be 

 obtained by any process of passing to the limit. 



J. Wll.l.ARI) ('.liiiis. 

 New Haven, Conn., November 29. 



Fof R I er's series arises in the attempt to express, by an in- 

 finite series of sines (and cosines) of multiples of .v, a function 

 uf .1 which has given values in an interval, say from .r = - ir 



NO. 1522, VOL. 59] 



to .v = Tr. There is no " curve " in the problem. Curves occur 

 in the solution of the problem, and there they occur by way 

 of illustration. There are two sorts of curves which occur. In 

 the first place, taking </> (jr) as the function to be expressed by 

 the series, andy(.i) as the sum of the series, we have the curves 

 y =0(.«) and y = f (.<), the graphs of the two functions. These 

 coincide wherever the series expresses the function ; but, if the 

 function <p(x} is one which cannot be expressed by a Fourier's 

 series for all values of r in the interval, the curves do not co- 

 incide throughout the interval. In the second place, taking 

 y„(.v) as the sum of the first n terms of the series, we have, the 

 family of curves y =f„{x), the graphs of /„(x) for difi'erent 

 values of «. As « increa.ses the graphs ol /'(.<) and /„(j) ap- 

 proach to coincidence in the sense that, if any particular value 

 of -I is taken, and any small distance (/ is specified, a number 

 «' may then be specified such that for every ;; greater than «', 

 the difference of the ordinates of the two curves is less than d. 

 But this is not the saine thing as saying that the curves tend 

 to coincide geometrically, and they do not in fact lie near 

 each other in the neighbourhood of a finite discontinuity of 

 ij)(.v). It is usual to illustrate the tendency to discontinuity of 

 /(.v) by noting the form of the curve j =y"„(.r) for large values 

 of«, but the shape of this curve always fails to give an indi- 

 cation of the sum of the .series for the particular values of .v for 

 which <^(x\ and / (.1 1 are discontinuous. This is the case in 

 the example cited by Prof. Willard Gibbs, where all particular 

 values between - ir and vr are equally indicated by the curve 

 y =y„(.v), but the sum of the series is precisely zero. 



Alay I point out that there is some ambiguity in the ex- 

 pression "the limiting form of the curve" used by Prof. 

 Willard Gibbs? Taking his example, it is quite true that h' 

 can be taken so great that, for every n greater than «', there is 

 a point of C„ within the given distance d of any point on the 

 broken line, but this statement is not quite complete. It is 

 also true that a number « can be taken great enough to bring 

 the point of C„ on any assigned ordinate within the given dis- 

 tance d of its ultimate position on the broken line, but it is 

 further essential to observe that no number « can be taken 

 great enough to bring every point of C„ within the given dis- 

 tance (/of its ultimate position on the broken line. The number 

 n which succeeds for any one ordinate always fails for some 

 other ordinate. Suppose, to fix ideas, that we take a point on 

 C„ for which y = \, and .v is nearly ir, so that ir .v is less than 

 d, and keeping r fixed, observe how y changes when n increases ; 

 it will be found that, for values of «/ very much greater than 

 «, the ordinate of C,„, for this x is very nearly ir, and «e can 

 in fact take /« great enough to make this ordinate lie between ir 

 and IT - d. In words, the representative point, which begins by 

 nearly coinciding with a point on a vertical part of the broken line, 

 creeps along the line, and ends by coinciding with a point on the 

 oblique part of the broken line. This will be the case for every 

 value of X, near v = ir, with the single exception of the value ». 

 Thus, in the pass.ige to the limit, every point near the vertical 

 part of the broken line disappears from the graph, except the 

 points on the axis of i . This peculiarity is always presented by 

 a series whose sum is discontinuous ; in the neighbourhood of 

 the discontinuity the series does not converge uniformly, or the 

 graph of the sum of the first n terms is always appreciably 

 different from the graph of the limit of the sum. 



In this way the graph of the sum of the first « terms fails to 

 indicate the behaviour of the function expressed by the limit of 

 this sum, and we m.ay illustrate the distinction between the two, 

 as Prof. Willard Giblis does, by considering the intersections of 

 the graph with lines parallel to the axis of .1. Keeping.j' fixed, 

 say^ = I, we may find, in his e.\ample, a number //, so that 

 there is a corresponding value of x differing from ir by less than 

 d, and then, allowing n to increase indefinitely, we shall get a 

 series of values of i , having ir as limiting value. Kul this limit- 

 ing value is not attained. In Prof. Willard (libbs's notation, 

 the equation zf„ (,v) = i has a root near to ir when « is great, 

 and « can be taken so great that the root ditTers from ir by less 

 than any assigned fraction ; but the equation 



limit 



2/„{.v) = I 



has no real root. In fact Prof. Willard Gibbs's " limiting f^rni 

 of the curve " corresponds to limits which are not attained : but 

 the limiting form in which the vertical portions of the broken 

 line are repKiced by the points where they cut the axis of .v 

 corresponds to limits which are effectively attained. It is the 



