May 31, 1906] 



NA TURE 



115 



•cross is as nearly as possible intermediate between a wild 

 iiorso and a Celtic pony. Of the skeleton it is, of course, 

 inipcssible to speak, but, judging by the shortness of the 

 trunk, the form of the head, and the conformation of the 

 limbs, the probability is that thiru are only five lumbar 

 vcricbrji:, as in the Moscow and St. Petersburg skeletons, 

 and that the skull and limb bones resemble those of a 

 young Prejvalsky horse. .After very full consideration, 

 .Salensky some years ago came to the conclusion that the 

 'larpan is a type specialised more to the side of E. cabalhis 

 than to E. prejvalskii. 



When all the facts now available are taken into con- 

 sideration, there seems no escape from the conclusion that 

 the Tarpan, once eonunon in the east of Europe, cannot 

 be considered as a true wild species. 



Further, it may be assumed that the Tarpan herds were 

 derived from at least three primitive stocks, viz. : — 

 ■(i) from a variety or species identical with or closely 

 related to the wild horse {£. prejvalskii) still surviving in 

 Central .Asia ; (2) from a variety having the characteristics 

 of the Celtic pony — E. c. celticus ; and (3) from a variety 

 a-esembling the forest horse — E. c. typicus. It is only by 

 assuming the multiplex origin of Tarpans that it is possible 

 ■to account for some of them having a heavy head, long 

 ears, a nearly upright mane, a mule-like tail, and five 

 Jumbar vertebrae, thus suggesting E. prejvalskii ; for 

 others, wanting the hind chestnuts and possessing a skull 

 like that of certain Scottish ponies, thus suggesting E. c. 

 ■celticus ; and for others having a thick head, full mane 

 and tail, and hind as well as front chestnuts, thus suggest- 

 ing E. c. typicus. 



Bv experiments now in hand I hope to settle what part 

 Prejvalsky's horse has taken in forming the Tarpan. If 

 I succeed in showing that crosses between Prejvalsky's 

 liorse and either the forest, Celtic, or Libyan variety are 

 ■practically Identical with the cross between the Shetland 

 mare and the Welsh pony stallion, I shall prove that at 

 least certain of the domesticated breeds are indebted to 

 Prejvalsky's horse for some of their characteristics, and at 

 the same time bring additional evidence in support of my 

 view that domesticated races have had a multiple origin, 

 and include plain as well as striped forms amongst their 

 less remote ancestors — have not, in fact, as Darwin 

 thought, descended from a single dun-coloured more or 

 Jess striped primitive stock. 



THE FIGURE AND STABILITY OF A LIQUID 

 SATELLITE.'^ 



"j\/r ORE than half a century ago Edouard Roche wrote 

 his celebrated paper on the form which a liquid 

 satellite will assume when revolving, without relati%-e 

 motion, about a solid planet." .As far as I know, his 

 laborious computations have never been repeated, and their 

 verification and extension form a portion of the work con- 

 tained in the present paper. 



Two problems involving almost identical analysis, but 

 very distinct principles, are here treated simultaneously. 

 If we imagine two detached masses of liquid to revolve 

 about one another in a circular orbit without relative 

 motion, the determination of the shapes of each of them 

 is common to both the problems ; it is in the conditions 

 of their secular stability, according to the suppositions 

 made, that the division occurs. 



The friction of the tides raised in each mass by the 

 attraction of the other is one cause of instability. If now 

 the larger of the two masses were rigid, whilst still possess- 

 ing the same shape as though liquid, the only tides subject 

 to friction would be those in the smaller body. It amounts 

 to exactly the same whether we consider the larger mass 

 to be rigid or whether we consider it to be liquid, and 

 agree to disregard the instability which might arise from 

 the tidal friction of the tides generated in it by the smaller 

 body. Accordingly I describe secular stability in the case 

 just considered as "partial," whilst complete secular 

 stability will involve the tidal friction in each mass. 



The determination of the figure and partial stability of a 



1 By Sir G. H. Darwin, K.C.B., F.R.S. Read before the Royal Society 

 on February 8. 



- Mt^m. Acad. Set. de Montpellier, vol. i., 1847-50, p. 243. 



NO. 1909, VOL. 74] 



liquid satellite is the problem of Roche. It is true that 

 he virtually considered the larger body or planet to be a 

 rigid sphere, but in this abstract the distinction intro- 

 duced by the fact that I treat the planet as ellipsoidal 

 may be passed over. It appears that, as we cause the two 

 masses to approach one another, the partial stability of 

 Roche's satellite first ceases to exist through the deform- 

 ation of its shape, and certain considerations are adduced 

 which show that the most interesting field of research is 

 comprised in the cases where the satellite ranges from 

 infinite smallness relatively to the planet to equality thereto. 



The limiting partial stability of a liquid satellite is deter- 

 mined by considering the angular momentum of the system, 

 exclusive of the rotational momentinn of the planet. This 

 corresponds to the exclusion of the tidal friction of the 

 tides raised in the planet. For any such given angular 

 momentum there are two solutions, if there is any. When 

 these two solutions coalesce for minimum angular momen- 

 tum, we have found a figure of bifurcation ; for any other 

 larger angular momentum one of the solutions belongs to 

 an unstable series and the other to a stable series of 

 figures. Thus, by determining the figure of minimum 

 partial angular momentum, we find the figure of limiting 

 partial stability. 



The only solution for which Roche gave a numerical 

 result was that in which the satellite is infinitesimal re- 

 latively to the planet. He found that the nearest possible 

 infinitesimal satellite (which is also in this case the satellite 

 of limiting partial stability) has a radius vector equal to 

 2.44 radii of its spherical planet. He showed the satellite 

 to have an ellipsoidal figure, and stated that its axes were 

 proportional to the numbers 1000, 496, 469. In the paper 

 the problem is solved by more accurate methods than those 

 used by Roche, and it is proved that the radius vector is 

 2-4553, 2"'' that the axes of the ellipsoid are proportional 

 to 10,000, 5114, 4827. The closeness with which his 

 numbers agree with these shows that he must have used 

 his graphical constructions with great care. 



For satellites of finite mass the satellite is no longer 

 ellipsoidal, and it becomes necessary to consider the de- 

 formation by various inequalities, which may be expressed 

 by means of ellipsoidal harmonic functions. 



The general effect for Roche's satellites of finite mass 

 in limiting partial stability is that the ellipsoidal form is 

 very nearly correct over most of the periphery of the 

 satellite, but at the extremity facing the planet there is a 

 tendency to push forth a protrusion towards the planet. 

 In the stable series of figures up to limiting stability this 

 protrusion is of no great magnitude, but in the unstable 

 series it would become strongly marked. When the un- 

 stable figure becomes much elongated, we find that it finally 

 overlaps the planet, but before this takes place the 

 approximation has become very imperfect. 



Turning now to the case of complete secular stability, 

 where the tidal friction in each mass is taken into account, 

 we find that for an infinitely small satellite limiting stability 

 occurs when the two masses are infinitely far apart. It 

 is clear that this must be the case, because a rotating 

 liquid planet will continue to repel its satellite so long as 

 it has any rotational momentum to transfer to orbital 

 momentum through the intervention of tidal friction. Thus 

 an infinitesimal satellite will be repelled to infinity before 

 it reaches the configuration of secular stability. .As the 

 mass of the satellite increases, the radius vector of limiting 

 stability decreases with great rapidity, and for two equal 

 masses, each constrainedly spherical, the configuration is 

 reached when the radius vector is 2-19 times the radius of 

 either body. 



When we pass to the case where each liquid mass is a 

 figure of equilibrium, the radius vector for limiting stability 

 is still infinite for the infinitely small satellite, and 

 diminishes rapidly for increasing mass of the satellite. 

 When the two masses are equal the radius vector of limit- 

 ing stability is 2-638 times the radius of a sphere the mass 

 of which is equal to the sum of the masses of the 

 two bodies. This radius vector is considerably greater 

 than that found in the case of the two spheres, for the 

 2-19 radii of either sphere, when expressed in the same 

 unit, is only 1-74. Thus the deformations of the two 

 masses forbid them to approach with stability so near as 

 when they were constrainedly spherical. 



