January 19, 1899] 



NA TURE 



271 



surface of the solid sphere, at which point there might be a 

 hole or door through which a small suspended magnet could be 

 introduced into the interesting field of force. 



Thos. Alexander. 

 Engineering School, Trinity College, Dublin, January 2. 



Fourier's Series. 

 Permit me to make a few remarks on the notes of Prof. 

 \Villard Gibbs and Mr. Love in NATURE of December 29, 1898. 

 Using Prof. Willard Gibbs's notation, write/.v„ for the series 



- if 



+ - sin «,v. 



and let C„ denote the curve whose equation is j = 2/1,,. 



Consider the two curves C„ and Qn^■\^ Then it is easily seen 

 that these curves intersect, when .v = any multiple of ir/« + 1 ; 

 and (omitting proof, so as not to cumber your pages with 

 calculation, which is not difficult) it will be found that one of 

 the curves passes through the extremities of the maximum 

 ■ordinates of the other, and the other through those of the 

 minimum ordinates of the first, in each case the curves cutting 

 •one another at an angle whose tangent is 2. Hence if d is the 

 distance along the ordinate of one of these points of intersection 

 from the slant part of the limiting broken line [y = .r), on one 

 of the curves the like distance will be less than d on one side of 

 the point, and greater than d on the other side. The statement 

 of Prof. W. Gibbs that, "if any small distance be first specified, 

 a number «' may be then specilied such that for every value of 11 

 greater than n' the distance of atiy point in d from the broken 

 line, and of any point in the broken line from C„, will be /ess 

 than the specified distance d,'' is therefore incorrect. It is 

 doubtless true that C,, 4-1 is, as a whole, nearer to the broken 

 line than C,„ but it is not true that every point in it is so. 



The above, in fact, shows, for a particular case, what Mr. 

 Love has remarked in more general terms in his note. 



I cannot follow Mr. Love in his remark — if, as I suppose from 

 his argument, it is intended to be general and not limited to the 

 particular illustration — when he says "Thus, in the passage to 

 the limit, every point near the vertical part of the broken line 

 disappears from the graph, except the points on the axis of .v." 



May we not as legitimately reason thus? The maximum 

 ordinate of C,, nearest to -r = ir is that for which x = mr/n + i. 

 There is a point P corresponding to a value of x between 

 ftir/n + I and ir, whose ordinate is any fixed fraction (the half, 

 say) of the above maximum. If now it be increased without 



limit, P will in the limit coincide with the point ( tt,~ \. Thus 



the vertical part of the broken line, in this way of arriving at the 

 limit, K'/// appear in the graph. 



Would it not be more correct to say that, when 11 is infinite, 

 the limiting curve has ordinates for the value .v = tt indeter- 

 minate within the limits - tt and t ? R. B. Havwaru. 



Shanklin, Isle of Wight, January 5. 



The Decrease of Swallows and Martins. 



Wh.\t an age of contradictions this is ! a statement is put 

 forth one day by some one who has apparently every reason to 

 be an authority, and it is contradicted the next day by some one 

 else who also appears 10 have good ground to support his 

 contradiction. 



Whom are we to believe ? 



Before preparing the paper on the decrease of the Hirundinidoe, 

 which I was privileged to read at the conference of the Society 

 for the Protection of Birds, I sought for, and obtained a large 

 amount of valuable information on the subject, and embodied 

 the chief details in my paper. 



The most useful contribution concerning the destruction of 

 small birds, including swallows, in Italy, was from the pen of 

 .Mr. W. J. Stillman, who was, until recently, the Times corre- 

 spondent in Rome, and who, in the course of a letter published 

 in the Times of August 23, 1S9S, wrote : — " Swallows are iieltcd 

 by the thousand as they come to the shores of Italy in their 

 northward migration, and are eaten as food. They are also 

 caught in quantities in the most cruel manner with artificial flies 

 and fish-hooks." [The italics are mine.] 



Another correspondent wrote to me personally, telling of the 



wholesale slaughter of bird^ in the neighbourhood of Florence, 



and in other parts of Italy, and although he does not, in his 



. letter, actually specify swallows and martins as amongst the 



: slain, yet it is not unreasonable to conclude they were amongst 



NO. 1525, VOL. 59] 



the many small birds which he has seen daily in Florence, piled 

 up four or five deep on flat barrows, some five feet by three, 

 and he also says he had been told that the nelters are the real 

 cause of the extraordinary absence of birds throughout Italy. 



For many years past great complaints have been made against 

 the French and Italians because of the great destruction of wild 

 bird-life in the two countries ; many people have protested 

 against it, including "Ouida," who is, I believe, a resident in 

 Italy, and who has on several occasions used her pen in defence 

 of the birds. 



My statements about the destruction of swallows in France 

 were based on reports issued by the Agricultural, and the 

 Zoological Society of France, the report of the latter body being 

 subscribed to by three observers who had made special inquiries 

 on the subject, and who wrote : — 



" In the springs of 1887 and 188S, hampers were addressed to 

 the naturalists of Paris containing dead swallows in the flesh, not 

 only by hundreds, but by thousands. One lot of these birds, 

 destined for the millinery trade, was spoilt owing to the im- 

 possibility of preserving them from putrefaction. These swallows 

 had been captured in the Department of the Bouches-du-Rhone 

 by means of three procedures — the net, by fish-hooks, and by 

 electric wire." 



In the report issued by the Agricultural Society of France, in 

 1894, it was stated on the authority of M. Rosier (delegate of the 

 Society of Agriculture of the Gironde), " that in his district, at 

 the season of their passage, there are killed every year more 

 than a million of the Hirondelles." 



The foregoing extracts will, I venture to think, prove that I 

 did not speak without my book when I partly ascribed the 

 decrease of the Hirundinida; in this country to the massacre of 

 such numbers of the family in France and Italy ; and as a 

 further proof that they do not come to our shores so abundantly 

 as formerly, I have received letters from the keepers of some 

 of our lighthouses, who report that during recent years they 

 have remarked a very great falling off in the numbers of these 

 birds at the time of the spring migration, and have wondered as 

 to the cause. 



In our own country the miin disturbing element is, without 

 a shadow of a doubt, to be found in the action of the house- 

 sparrow, who has constituted itself a most persistent enemy of 

 the swallows, especially the house-martin, whom it harries in 

 every direction, and has been in many instances the cause of 

 whole colonies of martins forsaking their old haunts. 



I cannot altogether agree with the theory that the absence of 

 the swallows in their old numbers is due to climatic changes, 

 or to the decrease of their insect food ; last summer this part of the 

 country was swarming with winged insect life, so much so that 

 some days the air seemed almost alive, and our rose and fruit 

 trees were smothered with them, but in this town and its im- 

 mediate neighbourhood swallows and martins were quite scarce, 

 although all the local circumstances are, one would think, 

 favourable for them in every way. 



If climatic changes are an affecting influence against the 

 coming of the swallows to this country, why are not other 

 spring migrants affected in the same way? i.e, the chiff'-chaff (a 

 much earlier immigrant than the swallow), the nightingale, the 

 willow-wren, or the wagtails and many other species, which 

 during the last few years have shown a tendency to increase. I 

 do think, however, that the atmospheric conditions of some of 

 our towns, especially where there are large factories and other 

 sources of noxious vapours, may be the means of keeping the 

 swallows away locally. 



My belief then is, that a very large number of the Hirun- 

 dinidre are prevented from coming to us by the slaughter 

 which awaits so many of them in the course of their journey 

 from their winter quarters ; and that when those who do survive 

 the perils of the way ultimately reach this land, they are inter- 

 fered with to such an extent by the sparrows that they are not 

 able to multiply so freely as they would do under more favour- 

 able conditions, and so their number is kept reduced from year 

 to year. 



I am still seeking for trustworthy information from personal 

 observers, and shall be most grateful for any communication 

 that may be sent to me at this address, especially as I propose 

 extending my paper and publishing it in the form of a pamphlet, 

 in which will be included extracts from the letters which have 

 come to me from all over the country. 



T. Herbert Allchi.n'. 



Esher, Bower Mount Rijad, Maidstone, January 7. 



