EQUILIBRIUM OP ROTATING MASSES OP FLUID. 
417 
concerned, be treated as spherical, and the tide-generating potential is given with 
sufficient accuracy by a single term of the second order of harmonics. As the masses 
are brought nearer to one another, this approximation ceases to be sufficient, terms of 
higher orders of harmonics become necessary to represent the potential adequately, 
and the departure from sphericity of each mass begins to exercise a sensible deforming 
influence on the other. 
When the departure from sphericity of one body produces a sensible deformation in 
the other, that deformation in its turn reacts on the first, and thus the actual figure 
assumed by either mass may be regarded as a deformation due to the primitive 
influence of the other mass, on which is superposed the sum of an infinite series of 
reflected deformations. 
But each mass is deformed, not only by the tidal action of the other, but also by its 
own rotation about an axis perpendicular to its orbit. The departure from sphericity 
of either body due to rotation also exercises an influence on the other, and thus there 
arises another infinite series of reflected deformations. It is shown in this paper how 
the summations of these two kinds of reflections are to be made by means of the 
solution of three sets of linear equations for the determination of three sets of 
coefficients. 
The first set of coefficients are augmenting factors, by which the tides of each order 
of harmonics are to be raised above the value which they would have if the perturbing 
mass were spherical. It appears that, the higher the order of harmonics, the more do 
these factors exceed unity. 
The second set of coefficients correspond to one part of the rotational effects. They 
appertain to terms of exactly the same form as the tidal terms, and in the final result 
the terms to which they apply become fused with the tidal terms. These terms are 
the zonal harmonics of the several orders with respect to the axis joining the centres 
of the two masses. 
The third set of coefficients correspond to the remainder of the rotational effect, and 
they appertain to a different kind of deformation. These deformations are represented 
by sectorial harmonics involving cos 2(f), where cf> is azimuth measured from the plane 
passing through the axis of rotation of the system and the centres of the two masses. 
That term of this set which is of the second order of harmonics, and which represents 
the ellipticity of either mass augmented by mutual influence, is the only term which 
is considerable, even when the two masses are very close together; but the existence 
of the other harmonic deformations of this class is interesting. We may say, then, 
that all the tides of either mass are augmented above the values which they would 
have if the other mass were spherical; that the ellipticity corresponding to rotation is 
augmented ; and that the deformation due to rotation is no longer exactly elliptic- 
spheroidal. 
The angular velocity of the system is found by the consideration that the repulsion 
due to centrifugal force between the two masses shall exactly balance the resultant 
MDCCCLXXXVII.-A. 3 H 
