584 



NA TURE 



(October ji, 1900 



action. The same thing will, of course, hold for a primary and 

 the atmosphere of a satellite. 



In the more general case, that in which the satellite has rota- 

 tional velocity a> about its axis (supposed perpendicular to the 

 plane of the orbit), we have, assuming that the satellite is 

 spherical and denoting o>-^,jffl by c, »il^l>{2 + c)a^lr^. This 

 agrees with the former result when c=l. These results were 

 first obtained by M. Roche. ^ 



The figure of a fluid satellite is determined by finding a surface 

 to which the resultant of the gravitational pull of the primary on 

 unit mass, a force «V equal and opposite to the gravitational 

 pull on unit mass at the centre, the gravitational force per unit 

 mass exerted by the matter of the satellite itself, and the centri- 

 fugal force of unit mass, is everywhere perpendicular. A first 

 approximation to the force due to the satellite itself is obtained 

 by neglecting the deviation from sphericity, as is done above. 

 But into this discussion we cannot here enter. It can only be 

 stated that the final result, taking into account the distortion of 

 the satellite, is that the satellite will be broken up if it approaches 

 closer to the primary than the limit given by the inequality 



r>2-44 



^•f.Ai- 



Now imagine a planet and a satellite moving round the sun, 

 the satellite being destitute of relative velocity. The satellite 

 may, for example, be regarded as a particle (of unit mass say) 

 of a ring of small mass composed of particles surrounding the 

 primary at a distance a, the whole turning, if that were possible, 

 with angular velocity equal to that of the primary round the sun. 

 By what we have seen above, the excess of solar attraction 

 over the sunward acceleration is, for a particle on the side 

 nearest the sun, ■^kyiajr^. This must be balanced by the 

 attraction kmla^, so that we have the equality kmla^ = ^iMa/r^. 

 From these expressions for the forces we see that the potential 

 energy, with a term for centrifugal force included, may be taken, 

 for unit mass of the infinitesimal satellite on the line of centres 

 at the point nearest the sun, as - km/a- -^kMa^/r^. This is an 

 example of the almost self-evident principle, known as the 

 theorem of Coriolis, that if there be included a term in the 

 potential energy which will give the components of centrifugal 

 force, we may write down the equation of relative kinetic 

 energy, just as if the rotating axes were fixed. The potential 

 energy thus required for the centrifugal force on a particle of 

 unit mass at a point at distances |, 77, ( from the planet's centre, 

 measured respectively along the line of centres, perpendicular 

 to this line in the plane of motion, and perpendicular to the 

 plane of motion, is -i«^{(^-|)2 + 7j-}, or, since «V = /('M/;'2, 



The total potential energy being taken as 



(with, as before, 



:kM/r^), the equation of relative kinetic energy is 

 I f> r, 2 r^ {\ V 



c^ 



and 



+ V' + 



r}]. 



which, for an infinitesimal satellite, is Hill's equation as eiven 

 above. 



For the moon, which has mass m' sensible in comparison with 

 the mass, m, of the earth, the first term in the square brackets 

 should be k{m + m')lp. 



It may be noticed by the dynamical student that if the above 

 expression for the potential energy be denoted by V, we have 

 not 1= - 9V/9|, &c., for the equations of motion, but 



av/ai, 



il=-dYldn. r 



av/ac 



A. Gray. 



I have found since the above was written that the same elementary view 

 Atm c 'i^i'"",'" given by Roche himself in his paper " Recherches sur les 

 Atmospheres des Cometes," Annales de lObservatoire, t. v. 1850. Perhaps 

 I maj- here direct attention to a valuable paper by Roche (which may, how- 

 en?i;i H "% ^"""^ "i"*^ ^^^" •" '■^f^'-'-^d to, be well known to astronomers). 

 rff^W^,^/."^'.""''- Constitution du Systems %oWn^," Mevt.de [Acad. 

 gonic rel'^ltches. ""'" ^'^"^ ^ ^^"^'"''' ^''''°""' °*" '^^ ^"'^°"'" '=°^'"°- 



NO. 161 5, VOL. 62] 



ANTELOPES AND THEIR RECOGNITION 

 MARKS. 

 'pHE Tragelaphine Antelopes hold a unique position amongst 

 the hollow-horned ruminants. No other group can show 

 species so sharply contrasted in size and build as the massive 

 eland rising over sixty inches at the withers, and the dainty 

 little bush-buck which falls short of half that height. Only the 

 Indian black buck amongst the gazelles can match the nylghaie 

 and nyala for diversity of sexual colouring ; and for elegance 

 of form, coupled with beauty of marking and grandeur of carriage, 

 the kudu is surpassed by no species of mammal. 



Apart from certain features presented by the skull and horns, 

 the affinity between the species here mentioned is attested by 

 the markings of the skin. On a ground-colour shading from slate 

 to chestnut are distributed certain white spots, patches or stripes, 

 which crop up so persistently in the different genera as to leave 

 no doubt they are a heritage from a common ancestor. A com- 

 parison between the skins of the existing species suggests that 

 this ancestor was coloured somewhat as follows :— Body and 

 head yellowish red ; flanks and hind-quarters striped with white ; 

 on the throat two white patches, one at each end ; one or two 

 spots on the cheeks, a V-shaped stripe between the eyes, a white 

 chin, a white upper lip ; legs paler on the inner side, quite white 

 at base close to chest and groin, and with two white spots on the 

 pasterns in front. 



Some or all of these markings have been inherited with 

 scarcely an exception by every known species of Tragelaphine. 

 Sometimes the spots on the head, sometimes the stripes on the 

 body, sometimes the patches on the throat are suppressed ; but 

 even in extreme cases of suppression, a spot here, a stripe there, 

 persists as a tell-tale sign of descent. The usefulness of char- 

 acters so constant may be taken for granted. The nature of 

 their usefulness has been discussed by both Wallace and Darwin ; 

 but so great is the discord between the opinions of these' 

 authorities that one cannot think both are right. 



Referring to the importance of special marks for recognition 

 where many species of nearly the same size and general form inhabit 

 the same region, Mr. Wallace says : "It is interesting to note that 

 these markings for recognition are very slightly developed in the 

 antelopes of the woods and marshes. . . . The wood-haunting 

 bosch-bok {T. sylvaticus) goes in pairs, and has hardly any 

 distinctive markings on its dusky chestnut coat, but the male 

 alone is horned. The large and handsome kudu frequents 

 brushwood, and its vertical white stripes are no doubt protective, 

 while its magnificent spiral horns afford easy recognition. The 

 eland, which is an inhabitant of the open plain, is uniformly 

 coloured, being sufficiently recognisable by its large size and 

 distinctive form ; but the Derbyan eland is a forest animal, and 

 has a protectively striped coat. In like manner, the fine Speke's 

 antelope, which lives entirely in the swamps and among reeds, 

 has pale vertical stripes on the sides (protective), with white 

 markings on face and breast for recognition" ("Darwinism" 

 p.. 220). 



It may be inferred from this passage that the interest attached 

 to the slight development of recognition marks in the antelopes 

 of the woods and marshes lies in the needlessness of such marks 

 for species living apart and not herding with others of the same 

 general size and form. If, however, there is no likelihood of 

 confusion, it is not quite clear from what species the horns of the 

 kudu serve to distinguish their owner, nor what significance in 

 this connection is to be attributed to the occurrence of horns 

 only in the male of the bosch-bok. Similarly, it is not 

 clear what use Speke's marsh-buck can have for recognition 

 marks. If, however, the spots on the face and throat subserve 

 recognition in this species, we must also conclude they are 

 retained for that purpose in the bongo ( T. euryceros), the lesser 

 kudu, the nyala (/'. angasi), in which they are very conspicuous, 

 as well as in the various smaller kinds of bush-buck, which in 

 other parts of Africa live the same life as the bosch-bok of the 

 Colony. Surely, too, Derby's eland is at least as recognisable by 

 its large size and distinctive form as the Cape species ; yet it is 

 adorned with a conspicuous V-shaped stripe between the eyes, 

 and the lower throat-patch forms a white collar, standing boldly 

 out against the black hue of the neck. 



In short, if the marks in question have been preserved for 

 recognition, it is singular that they are exceptionally well de- 

 veloped in the species that live in pairs or small parties by them- 

 selves in thick bush — species which, according to the hypothesis, 

 have little, if any, need of them. It is conceded, of course, that 

 the spots on the head and throat, like the stripes on the body, -4 



