August 1 1, 1887] 



NATURE 



359 



a deformation due to the influence of the other considered as a 

 sphere, on which is superposed the sum of an infinite series of 

 deformations of each due to the deformation of the other and of 

 itself. 



But each mass is deformed, not only by the tidal action of 

 the other, but also by its own rotation about an axis perpen- 

 dicular to its orbit. The departure from sphericity of either 

 body due to rotation also exercises an influence on the other 

 and on itself, and thus there arises another infinite series of 

 deformations. 



It is shown in the paper how the summations of these two 

 kinds of reflected influences are to be made, by means of 

 the solution of certain linear equations for finding three sets of 

 coefiicients. 



The first set of coefficients are augmenting factors, by which 

 the tide of each order of harmonics is to be raised above the 

 value which it would have if the perturbing mass were spherical. 

 The second set c >rrespond to one part of the rotational effect, 

 and belong to terms of exactly the same form as the tidal terms, 

 with which they ultimately fuse. The third set correspond to 

 the rest of the rotational eftect, and appertain to a diff"erent 

 class of deformation, which are in fact sectorial harmonics of 

 difierent orders. The term of the second order represents the 

 ellipticity of the mass due to rotation, augmented, however, by 

 mutual influence. AH the terms of this class, except the second, 

 are very small ; their existence is, however, interesting. 



From the CO isideration that the repulsion due to centrifugal 

 force shall exactly balance the attraction between the two masses, 

 the angular velocity of the system is found. It is greater than 

 would be the case if the masses were spherical. 



The theory here sketched is applied in the paper numerically, 

 and illustrated graphically in several cases. 



When the masses are equal to one another they are found to 

 be shaped like flattened eggs, and the two small ends face one 

 another. Two figures are given, in one of which the small 

 ends nearly touch, and in the other where they actually cross. 

 In the latter case, as two portions of matter cannot occupy the 

 same space, the reality must consist of a single mass of fluid 

 consisting of two bulbs joined by a neck, somewhat like a 

 dumb-bell. In the figure conjectural lines are inserted to show 

 how the overlapping of the masses must be replaced by the neck 

 of fluid. 



Fig. I 



Section perpendicular to .axis of rotation. 



Section through axis of rotation. 



A comparison is also made between the Jacobian ellipsoid of 



uilibrium with three unequal axes and the dumb-bell. It 



jiears that with the same moment of momentum the angular 



velocity is nea'-ly the same in the two figures, but the kinetic 



!'nergy is a little less in the dumb-bell. The intrinsic energy of 



dumb-bell is, however, greater than that of the ellipsoid, so 



t the total energy of the dumb-bell is slightly greater than 



t of the ellipsoid. 



Sir William Thomson has remarked on the "gap between 



a: unstable Jacobian ellipsoid and the case of the 



mallest moment of momentum consistent with stability in two 

 :qual detached portions." "The consideration," he says, "of 

 how to fill up this gap with intermediate figures is a most attrac- 

 ive question, towards answering which we at present offer no 



contribution." ^ This paper is intended to be such a contribu- 

 tion, although an imperfect one. 



M. Poincare has made an admirable investigation of the forms 

 of equilibrium of a single rotating mas, of fluid, and has spe- 

 cially considered the stability of Jacobi's ellipsoid." He has 

 shown by a difficult analytical process, that when the ellipsoid 

 is moderately elongated, instability sets in by a furrowing of the 

 ellipsoid along a line which lies in a plane perpendicular to the 

 longest axis. It is, however, extremely remarkable that the 

 furrow is not symmetrical with respect to the two ends, and 

 there thus appears to be a tendency to form a dumb-bell with 

 unequal bulbs. 



M. Poincare's work seemed so important that, although the 

 figures above referred to were already drawn a year ago, this 

 paper was kept back in order that an endeavour might be 

 made to apply the principles enounced by him, concerning 

 the stability of such systems. The attempt, which proved 

 abortive on account of the imperfection of approximation of 

 spherical harmonic analysis, is given in the appendix to the 

 paper, because, notwithstanding its failure, it presents features 

 of interest. 



The calculations in this paper being made by means of 

 spherical harmonic analysis, it is necessary to consider whether 

 this approximate method has not been pushed too far in the 

 computation of figures of equilibrium which depart consider- 

 ably from spheres. A rough criterion of the applicability of the 

 analysis is derived from a comparison between the two values of 

 the ellipticity of an isolated revolutional ellipsoid of equilibrium 

 as derived from the rigorous formula and from spherical har- 

 monic analysis. As judged by this criterion, which is necessarily 

 in some respects too severe, the figures drawn appear to present 

 a fair approximation to accuracy. 



Since, as above stated, the rigorous method of dis:ussing the 

 stability of the system fails, certain considerations are adduced 

 which bear on the conditions under which there is a form of 

 equilibrium consisting of two fluid masses in close proximity, 

 and it appears that there cannot be such a form, unless the 

 smaller of the two masses exceeds about one-thirtieth of the 

 larger. It seemed therefore worth whilfe to find to what results 

 the analysis would lead when two masses, one of which is 

 twenty-seven times as great as the other, are brought close 

 together. As judged by this criterion the computed result must 

 be very far from the truth, but as the criterion is too severe, it 

 seemed worth while to give the figure. The smaller mass is 

 found to be deeply furrowed la a plane parallel to the axis of 



Fig. 2. 



Ratio of masses i : 

 rotation. 



[7. Upper iialf of figures section through axis of 

 Lower half section perpendicular to axij. 



rotation, so as to be shaped like a dumb-bell, and although this 

 result can only be taken to represent the truth very roughly, yet 

 it cannot be entirely explained by the imperfection of the 

 analytical method employed. It appears then as if the smaller 

 body were on the point of separating into two masses, in the 

 same sort of way that the Jacobian ellipsoid may be traced 

 through the dumb-bell shape until it becomes two masses. 



M. Poincare has commented in his paper on the possibility of 

 the application of his results, so as to throw light orrthe genesis 

 of a satellite according to the nebular hypothesis, and this 



' Thomson and Tail, "Natural Philosophy" (1883), § 778 (i). He also 

 remarks elsewhere that by thinning a Jacobian ellipsoid in the middle, we 

 shall get a figure of the same m im ;nt of m :)mentum and less kinetic energy. 



« Acta Math. vii. 3 and 4, 1885. 



