October 



1898] 



NA TURE 



569 



LETTERS ^ TO THE EDITOR 



[ The Editor does vol hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

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

 manuscripts intended for this or any other part of NATURE. 

 No notice is taken of anonytnous communications. 1 



The Aurora Borealis of September 9. 



I HAVE read, with much interest, in Nature of September 

 15, the article concerning the aurora borealis of September 9, 

 and it may be of interest to your readers to know that this 



but very bright streamers in all directions, especially to the 

 east. This latter formation was surrounded by quite black 

 spaces of sky, which made the luminous phenomena look more 

 beautiful. 



Meanwhile, in the northern part of the sky, the aurora took 

 the shape of ever-changing columns, and long, sometimes spiral 

 and undulating bands, which twice, in the north-west and in the 

 north east, doubled, resembling curtains hanging one over the 

 other. 



A little after eleven I saw in the north a very strange form- 

 ation of aurora ; three vertical columns in their upper part were 

 crossed by a bright horizontal streamer, extending nearly from 

 north-west to north-east. 



Soon after 11.30 the aurora began to 

 vanish everywhere, and, in a very marked 

 manner, took more and more the aspect of 

 some luminous shapeless cloud. After 12 

 o'clock all traces of columns and streamers 

 disappeared, and at I o'clock nothing 

 more of the phenomenon was to be seen. 

 N. Kaulbars. 

 Helsingfors, September 28. 



beautiful phenomenon displayed its splendours the same evening 

 in all parts of Finland territory. 



On that day I had the good fortune to see it in Helsingfors, 

 from its earliest beginning to its end, in a clear, perfectly 

 cloudless sky, and a calm and transparent air. These favourable 

 conditions enabled me to sketch the principal movements of it, 

 and I send you herewith a copy of the drawing I made. 



The aurora was not only one of the most splendid that has 

 been seen, but also that has appeared in our latitude for a long 

 series of years. It began a little before 9 o'clock, and at 10 

 arrived at its maximum brilliancy, a state in which it, ever 

 changing, remained till 1 1 o'clock, display- 

 ing the whole time an exceedingly beautiful 

 brightness in all its parts. 



The display began with a very bright 

 arc in the north, but this very soon disap- 

 peared, while at the same moment exceed- 

 ingly brilliant streamers extended at once 

 up from the western and eastern horizons, 

 sending immense columns to the zenith; 

 and taking the shape of a colossal arc 

 arching the whole sky from horizon to 

 horizon. Masses of light flowed from both 

 sides to the zenith, where they seemed to 

 disappear. At 10 o'clock the great arc 

 was interrupted on both sides by a dark 

 region, the bright streamers remaining 

 only on opposite horizons ; but in the 

 same moment a corona of the highest 

 splendour appeared in the zenith, consist- 

 ing of three nearly parallel streamers, 

 stretching from west to east, and ending 

 towards tiie west in the dark space, and 

 towards the east in a beautiful fan of light. 

 Half an hour later the corona took the 



shape of an immense dome, the ribs and columns of which 

 stood around all parts of the horizon. The whole visible sky 

 at that moment presented one single enormous dome of in- 

 describable beauty. The brightest columns of this dome were 

 to the west and to the east, those to the north were much less 

 bright, and the columns to the south were scarcely visible. 

 From every part of this dome streamers of light, without 

 interruption, flowed up to the zenith. 



At II o'clock, when the dome suddenly disappeared, the 

 corona took the shape of a luminous spiral-ring, sending short 



NO. 151 I, VOL. 58] 



Fourier's Series. 

 In a letter to Nature of October 6, 

 Prof. Michelson, referring to the statement 

 that a Fourier's series can represent a 

 discontinuous function, describes " the idea 

 that a real discontinuity can replace a sum 

 of continuous curves" as "utterly at vari- 

 ance with the physicists' notions of quan- 

 tity." If, as this seems to imply, there are 

 physicists who hold " notions of quantity " 

 opposed to the mathematical result that 

 the sum of an infinite series of continuous 

 functions may itself be discontinuous, they would be likely to profit 

 by reading some standard treatise dealing with the theory of 

 infinite series, such, for example, as Hobson's "Trigonometry," 

 and the paper by Sir G. Stokes quoted on p. 251 of that work. 



Prof. Michelson takes a particular case. He appears to find 

 a difficulty in the result that the sum of the series 



>» = 2[sin jr - i sin 2j: + i sin 3jr - ...] 



is equal to x when x lies between - it and ir, is equal to 

 - 2ir -f jr when x lies between ir and 3ir, and so on, and further 

 is equal to zero when x is - ir, or ir, or 3ir, and so on. 



With the view of stating his difficulty simply, he has tried to 

 sum this series, and the series obtained from it by diflferentiating 

 its terms, for values of x of the form ir -t- «, where it appears to 

 be meant that c is positive and less than 2w. 



The series (thus obtained) for y and dyldx are given by the 

 equations 



,dy 



-f-4sin2« -t- isin 3* -H ... + i«I»«« + 



\.d. = cos « + cos 2€ -I- cos 3« -f ... + cos «« 

 dx 



