126 



NA TURE 



[June 9, 1892 



point out to the charitably disposed that there are 



a number of desiderata : there are, for example, no 



specimens of either the African or the American 

 " Fin-foots." 



LETTERS TO THE EDITOR. 



[TAe 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 iiATVKV., 

 No notice is taken of anonymous communications. ] 



The Line Spectra of the Elements. 



The proper replies to Prof. Runge's letter in last week's issue 

 of Nature are three in number: viz. (i) that, as I pointed 

 out in my former letter (Nature of May 12, p. 29), the reason- 

 ing in my paper is valid if, as I there proved and as Prof. 

 Runge now admits in the first sentence of his letter, Fourier's 

 theorem can be applied to motions which approximate to non- 

 periodic motions in any assigned degree and for any assigned 

 time ; (2) that I am not aware of anything I have written 

 which countenances Prof. Runge's supposition that " Prof. 

 Stoney has not noticed that a distinct property of the 

 function is wanted in order to get a proper " [rather, a mathe- 

 matically accurate] "resolution into a sum of circular functions " ; 

 and (3) that Prof Runge is mistaken when he supposes 

 that "the amplitudes and periods" [or frequencies] " of the 

 single terms ... do not approach definite values when the 

 interval" [i.e. the periodic time of the recurrence required 

 by Fourier's theorem] "increases indefinitely." 



What the true state of the case is, is most easily shown as 

 regards the frequencies of the lines ; and as the proof is, I 

 believe, new, and leads to a result of importance in the inter- 

 pretation of spectra, I subjoin it. 



Take a motion of the electron — 



X = The sum of partials such as ( . 



2Tkt , , 2lTkt\ I . 



a sm — :— + b cos — :— ); (i) 

 J J ' 



with similar expressions for the other two co-ordinates ; in 

 which the oscillation-frequencies, the ^''s, may be commensur- 

 able with one another, or incommensurable. If incommensur- 

 able, the motion is non-recurrent. Let this motion be arrested 

 at intervals of T, and immediately started afresh as at the be- 

 ginning. We thus obtain a recurrent motion consisting of a 

 certain section of the motion (i) repeated over and over again. 

 This new motion can be analyzed by Fourier's theorem, and we 

 have to inquire what we thus obtain. Without losing anything 

 in generality, we may confine our attention to the motion 

 parallel to the axis of x, and to the single partial of that motion 

 which is written out above, as all the partials lead to similar 

 results. 



Let us then examine by Fourier's method the motion which is 

 represented by the equation — 



. 2irkt , 2irkt , , 



Xi: = a sin — .^ -t- b cos — .- (2, a) 



J J 



from t — o till t = T, and which is repeated from that on at 



intervals of T. If T is a multiple of j/k, Fourier's theorem 



simply furnishes equation (2, a) as the complete expression for 



all time of the motion ; so that in this case it indicates the same 



definite line in the spectrum as is furnished by the original 



partial of equation (i). 



If T is not a multiple ofj/k, 



T will = (;« + a)4 , 

 k 



where m is a whole number and 

 (2, a) then becomes 



2Tr{m + a)t 

 t 



which is true from t = otUlt = T, after which the motion is 

 to be repeated. Then, by Fourier's theorem — 



xa 



proper fraction. 

 '> cos ?l(^L±^_^ 



Equation 

 • . (2, b) 



j;/: = Ao -f Aj sin ~ 



-fB,sin?!^' 

 T 



NO. II 80, VOL. 46] 



A, sin ^^ 4- 

 B,sin^'-h 



(3) 



is true of this motion for all time, in which 



T T 



A„ . j sin^ ^Jlf.dt = a sin ^_^^(^1±A' . sin ^-^Z . dt 

 + ^/'cos^-^<'"^,"-«l'.sin^'r'.^/; 



^T T 



■D / o 2irnt ,, I . 27r(w -f a)t 2irnt ,, 



D„ . I cos- -^^_ . or = rt / sm ^ ^ cos --- . dt 



, ; / 27r(w 4- a)t ^^^ 2TTnt ,, 



+ b j cos -^ . cos — --, - .dt ; 



which, when integrated, give the following values for A« and 

 B„— 



. _asin27ra - b{l - cos 27ro) 



G-") 



■D _ a( I - cos 2ira) + h sin 2ira/ l^ 

 27r \7l 



.(4) 



where d stands for {m - n + a), and s for {;« + n + a). 



This furnishes a very remarkable spectrum, a spectrum of lines 

 that are equidistant on a map of o-;cillation-frequencies, and that 

 extend over the whole spectrum. But they are of very 

 unequal intensities. If T is a long period, m is a high 

 number. The lines are then ruled close to one another, and 

 their intensities are insensible except when n is nearly equal 

 to m, the two brightest Hnes being the next to the position of 

 the original line of equation (i), one on either side of it, and the 

 others falling off rapidly in brightness in both directions. 



If we take a longer period for T, w becomes a still higher 

 number ; the lines are more closely ruled and are more suddenly 

 bright up to those on either side of the position of the original 

 line of equation (i), to which also they are now closer ; so that, 

 at the limit, when T increases indefinitely, equation (3) becomes 

 a mathematical representation of the original line of equation (i). 



This interesting investigation is all the more important as it 

 gives a clue to how rulings of lines which are equidistant and 

 brighter up to the middle may arise ; and I feel sure that Prof. 

 Runge will join me in not regretting that he expressed the 

 doubts which led to its solution. 



G. Johnstone Stoney. 



9 Palmerston Park, Dublin, June 3. 



Stone Circles, the Sun, and the Stars. 



Articles by Mr. Norman Lockyer and Mr. Penrose, re- 

 cently puldished in Nature, have dealt with the positions of 

 ancient Egyptian and Greek temples with relation to the rising 

 sun, and to the pole star, or some star or stars in its vicinity. 

 For some years past I have endeavoured to show, in papers read 

 before the British Association and other Societies, that our stone 

 circles had a relation to the rising sun, indicated usually by an 

 outlying stone or by a notable hill-top in the direction in which 

 the sunrise would be seen from the circle, and I have in some 

 cases found similar indications towards the north, which may 

 have referred to the pole or other northern star or stars. A 

 paper containing many details as to these cases will shortly 

 appear in the Journal of the Royal Archaeological Insthute. 



There are six circles on Bodmin Moors, which at first sight 

 appear to have no relation to each other, but which, if the 

 6-inch Ordnance map is to be relied upon, would seem to have 

 been arranged on a definite plan (see accompanying plan). 



The Stannon and Fernacre Circles are in line i' (true) north 

 of east with the highest point of Brown Willy, the highest hill 

 in Cornwall ; and the Stripple Stones and Fernacre Circles are 

 in line with the summits of Garrow and Rough Tor, at right 

 angles with the other line— namely, 1° west of (true) north. A 

 line from theTrippet Stones Circle to the summit of Rough Tor 

 would also pass through the centre of one of the Leaze Circles 

 (about 12° east from true north). Other hills are in the direc- 

 tion of the rising sun. TheTrippet Stones are ii^° south of 

 west from the Stripple Stones, 10° east of south from the 

 Stannon Circle, and about 13° west of south from the Fern- 

 acre Circle. The respective bearings of the other circles have 

 already been given, and all are true (not magnetic) according to 

 the 6-inch Ordnance map. 



