May 6, 1897] 



NATURE 



I append a short proof, out of several which may be given. 

 We have to show that 



(ti'o -f~ ti'a)-"-! + (zc4 + K'j)'"-! = I mod 2n - i 



when, and when only in - i is prime. 



Let w.,, Tt'a, tc'4, K'5 be the unreal roots of .v' + i = o and 

 2M - I = any odd prime, then we may say 



'5 V 



v'5' 



I mod/ 



is any odd prime. 



= — ^'^ = I mod /or /, = I mod /. 



2P 



Now, by r'ermat"s theorem 2'*' 

 only/ is jirime. Thus 



/.;/ when, and when 



f.in - p.ti— o mod/ 



which proves the theorem for any odd prime. 

 It is also true for/ = 2, since by ordinary work 



- 5 



2 n/5 + S 



= 3=1 mod 2. 



Thus the theorem is universally true for all primes. 



It is remarkable that the second factor of the prime series 

 given above is also a function of the prime /, viz. : 



i,,/~3+/-4./-5^/-5./-6 P-l^^^^_ 



X. gr. the 4th term of tlie prime series is 29, thus 



7-4 • 7-5 ■ ^ 

 -, ! / 



7 ;i -f 2 + i; = 28. 



As this communication is somewhat long, I reserve the proof 

 of this. RoBT. W. D. Christie. 



Ajjril 28. 



The Effect of Sunlight on the Tints of Birds' Eggs. 

 The beautiful and delicate colours observed on the eggs of 

 birds are not very fast to light, more especially when they 

 belong to the lighter class of colours. Egg-collections should 

 be carefully protected from the light by some covering over the 

 case, when they are not being inspected ; otherwise much of 

 their beauty of tint becomes lost in course of time. It is gratify- 

 ing to notice that in museums and natural history collections 

 this precaution of protecting egg-cases with covers is now 

 almost universally observed. In many instances some of the 

 finest and most characteristic tints of several eggs disappear on 

 exposure to much sunlight. A common example may -be found 

 in the beautiful pale blue of the starling's Qgg(S/urnus vulgaris). 

 Thi.s, on exposure to sunlight for a few days, loses its clear blue- 

 ness of tone, and becomes purplier, approaching more to the 

 slate tint. Such is also the case with most of the greenish -blue 

 eggs, like those of many sea-birds, the common guillemot's 

 ( Uria troile), for instance, the beauty of which largely depends 

 on the clear freshness of its blue tints. The writer, some time 

 ago, made some experiments on the fastness to sunlight of 

 those egg-tints. The method employed was a very simple one, 

 and may be briefly described as follows. Various birds' eggs 

 were selected for experiment, those having decided and well- 

 marked colours being preferred. These shells were halved 

 lengthwi.se, care being taken before the operation to divide it so 

 that each half should, as nearly as possible, present the same 

 amount of colouring. One half was kept from the light for 

 future comparison, while the other half was exposed in a glass 

 case to direct sunshine. After various exposures, amounting to 

 one hundred hours' sunshine, each exposed half was then com- 

 pared with its unexposed counterpart, and the changes in hue 

 carefully noted. Little change was visible in the darker coloured 

 eggs of the olive-brown or chocolate depth, but in the lighter 

 tints, especially among the blues and green-blues, the changes 

 became more marked. Among the darker shades of eggs was 

 the common curlew's or whaup {Nunienitts arquata), with its 

 dull olive-green spotted with deep shades of brown ; and also 

 the lapwing {Vauellus cistatus), which closely resembles in 



NO. 1436. VOL. 56] 



general appearance that of the curlew. Such deeply-colouted 

 eggs are little altered on exposure to light, unless after very long 

 exposure, when they lose some of their rich warmth of ton^ 

 and become a trifle clearer in their ground tints, making them 

 look somewhat bleached. Many sea-birds' eggs have a bluish- 

 green colour— sea-green it might be called— which, when new 

 and unexposed, is rich and beautiful. This clear tint, however, 

 is lost on exposure, and it assumes a more dingy slate hue! 

 Some of their eggs have a network of white chalk-like incrusta- 

 tion streaked over the bluish ground tint. This may be seen on 

 the egg of the common cormorant {Phalacrocorax car ho). If 

 such shells be exposed for several days to sunlight, and after- 

 wards the white incrustation removed with a knife, the difference 

 produced on the ground tint . by exposure becomes at once 

 apparent. The exposed parts will be found of a slaty, duller hue, 

 more approaching a stone-grey tint ; while the unexposed parts, 

 protected by the incrustation, will reveal the original sea-green 

 tint in all its freshness. Another example is the fair blue egg 

 of the common thrush or mavis ( Turdtts mitstcus). This egg 

 when newly laid is of soft light blue of a fine shade, but on 

 exposure it loses much of this clearness of tint, and becomes 

 dull and purply, tending more to a leaden hue. Many similar 

 examples might be given of beautiful shades of blue and blue- 

 green tinted eggs which all tend to become redder and duller on 

 exposure. The red blotched egg of the fieldfare ( Tiirdus pilaris) 

 fades in this manner, and the red markings assume a lighter 

 rusty-brown hue. The ring owm\{Turdus tonpiatits) so well 

 known for its predatory visits to the strawberry-beds, has an 

 egg closely resembling the fieldfare's, both in ground tint and 

 markings, which undergoes the same changes in every respect. 

 One of the commonest eggs is that of the blackbird ; it also 

 loses its greenish hue and becomes more of a stone-grey, while 

 its varied markings lose considerably in depth. In the beautiful 

 eggs of the yellow hammer (Emberiza citriitella), so curiously 

 veined and mottled with dark red-brown over a pale ground, 

 little or no fading was visii)le after exposure. Its markings 

 may thus be considered fast to light. There are but few 

 coloured eggs which show no appreciable change after so severe 

 an exposure test as 100 hours' direct sunlight. A good example 

 of a fairly fast-coloured egg is that of the favourite songster 

 the skylark {Alaiida an<ensis). Its eggs vary considerably 

 in colour, but they are always of an indescribable hue, 

 sometimes an ashy brown, or a dark purplish grey, other 

 times more of a greenish tinge. These stand the light very 

 well. The specimens tested looked only a trifle bleached, 

 but those having the greener tinge fade more. One of the 

 prettiest of blue eggs is that of the the common hedge-sparrow. 

 The loss of its clear blue tint to a purplish blue drab was most 

 marked. To illustrate the unstable nature of egg-colouring in 

 comparison with colours of different origin, various other colours 

 resembling in tint those of the eggs were exposed in a similar 

 manner. These were " distemper " colours, and water colours, 

 painted on paper, and coal-tar colours dyed on wool. The 

 distemper colours were perfectly fast to light ; their colour con- 

 stituents all being of mineral origin. The water colours 

 examined were both of mineral and vegetable origin ; those 

 belonging to the latter faded very considerably. The coal-tar 

 colours selected were mostly of the bluish cast, corresponding to 

 many of the egg tints. The summary of the results obtained 

 might be tabulated as follows : — 



Colours examined. ^«""" ^f'^"" '°° '^°"'"'^" 



Distemper colours. . . 

 Water colours 

 Coal-tar colours ... ... ... 30 ,, ,, 



Egg-shell colours 20 ,, ,, 



The above results, along with the few common examples 

 which have just been given, readily show that eggs lose much 

 of their delicate and characteristic beauty of tint on being too 

 freely exposed to sunlight. David I'aterson. 



Rosslyn, Midlothian 



Physiological Specific Characters. 



Prof. R. MEtnoi-A, in his very suggestive presidential ad- 

 dress to the Entomological Society, remarks {Trans. Ent. Soc. 

 for 1896, Pt. V. p. Ixxviii.): — "At any rate, it appears to me 

 inconceivable that any change of environment requiring a modi- 

 fication of structure of sufficient magnitude to rank as diagnostic 

 in the systematic sense, should not also be accompanied by a 



exposure. 

 100 per cent. fast. 

 60 ,, 



