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 (%) for the sum of the first n terms of the series 



sin cc \ sin 2x + J sin 3x % sin 4a?-fetc. 



I suppose that there is no question concerning the form of the curve 

 defined by any equation of the form 



2/ = 2/ n (a). 



Let us call such a curve 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 (TT, TT), thence vertically in a straight 

 line to the point (TT, TT), thence obliquely in a straight line to the 

 point (3-7T, TT), 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 ri may be then specified, 

 such that for every value of n greater than ri, the distance of any 

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

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



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

 equation = limit 2/ n (a?). 



n= 



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 C n 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 C n with horizontal transversals, and 



