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Prof. Gr. H. Darwin. On Figures of [June 16, 



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 coefficients. 



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 correspond 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 rota- 

 tional effect, and appertain to a different class of deformation, which 

 are in fact sectorial harmonics of different orders. The term of the 

 second order represents the ellipticity of the mass due to rotation, 

 augmented, however, by mutual influence. All the terms of this 

 class, except the second, are very small; their existence is, however, 

 interesting. 



From the consideration 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. 



A comparison is also made between the Jacobian ellipsoid of equili- 

 brium with three unequal axes and the dumb-bell. It appears that 

 with the same moment of momentum the angular velocity is nearly 

 the same in the two figures, but the kinetic energy is a little less in 

 the dumb-bell. The intrinsic energy of the dumb-bell is, however, 

 greater than that of the ellipsoid, so that the total energy of the 

 dumb-bell is slightly greater than that of the ellipsoid. 



Sir William Thomson has remarked on the " gap between the 



unstable Jacobian ellipsoid and the case of the smallest 



moment of momentum consistent with stability in two equal 

 detached portions." " The consideration," he says, " of how to fill 

 up this gap with intermediate figures is a most attractive question, 



