418 
PROFESSOR G. H. DARWIN ON FIGURES OF 
attraction between them. If the masses were spherical, the square of the orbital 
angular velocity, multiplied by the cube of the distance between the centres, would be 
equal to the sum of the masses. When the masses are deformed, however, this law is no 
longer true, and the angular velocity has to be augmented by a factor a little greater 
than unity, which depends on the amounts of the deformations. 
The theory here sketched is applied above numerically to several cases, and the 
results will be found in the preceding paragraphs. We shall first consider the cases 
where the two masses are equal to one another. 
In the first example (/3 = y) solved numerically, the distance between the centres 
of the two masses is 2‘83 times the mean radius of either of them. The two bodies 
are found to be elongated until they approach near to one another; but, as the 
character of the distortion is better illustrated in a subsequent case, the result is not 
given graphically. All the data, however, are found which will enable the reader to 
draw the figure if he should wish to do so. 
In the next example (/3 = ^), with the masses still equal, the distance between the 
centres is reduced to 2’646 of the mean radius of either. The result of the solution is 
illustrated by two figures. In fig. 4, Plate 22, the section of the masses by a plane 
perpendicular to the axis of rotation is shown, and in fig. 5, Plate 23, we have the 
section by a plane passing through the axis and the centres of the two masses. On 
both figures are inscribed the values of the radii for each 15° of latitude in terms of 
the mean radius as unity, and the mean sphere, from which the distortion is computed, 
is marked by a short line on each radius. The elongation of the masses is, of course, 
considerably greater in the section through the axis than in the other section. Each 
mass is shaped somewhat like an egg, and the small ends face one another and come 
very nearly into contact. 
In the headings to the figures, amongst other numerical data, are given the square 
of the angular velocity and the angular momentum of the system. The density of 
the fluid being unity, the angular velocity a> is given by the value of orjiir ; this is 
the function of angular velocity which is usually given when reference is made to figures 
of equilibrium of rotating fluid, such as the revolutional or Jacobian ellipsoids of 
equilibrium. The moment of momentum of the system is given by reference to the 
angular velocity of a sphere, of the same mass as the sum of our two masses, rotating 
so as to have the same momentum. If, in fact, b be such a length that a sphere of 
fluid of that radius has the same mass as our system (so that b 3 = a 3 + A 3 ), then the 
moment of momentum is given by a number p in the expression (f-77-) 3 b° X p. By this 
notation the angular velocity and moment of momentum are made comparable with 
the results given in a previous paper'” on the Jacobian ellipsoid of equilibrium. 
From that paper the following table of the axes, angular velocity, and moment of 
momentum of several solutions of Jacobi’s problem is extracted. 
* ‘ Roy. Soc. Proc.\ vol. 41, 1S87, p. 319. 
