FOURIER'S SERIES. 259 



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 



and proceed to the limit for n oo , we do not necessarily get y for 

 x TT. We may get that ratio by first setting x = TT, and then passing 

 to the limit. We may also get y = ~L, X = TT, by first setting y = l, 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( since ~ sin 2cc... + sin nx) (1) 



\ 2 ~n / 



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 X, extend beyond the points where 

 they meet the inclined portions, their total lengths being expressed 

 by four times the definite integral 



f 



Jo 



smu 7 

 du. 



u 



If we call this combination of inclined and vertical lines C, and 

 the graph of equation (1) O n , and if any finite distance d be specified, 

 and we take for n any number greater than 100 /d 2 , the distance of 

 every point in C n from C is less than d, and the distance of every 

 point in C from C n is also less than d. We may therefore call C the 

 limit (or limiting form) of the sequence of curves of which 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. 



