406 
PROFESSOR G. H. DARWIN ON FIGURES OF 
Figs. 2 and 3 refer to (3 = 4 and 5 to that of /3 = and the numerical values for 
/3 = y, given above, make it easy to draw a figure for /3 = y. 
Since in these cases the masses are equal, the two halves of the figure are the 
images of one another. The numerical value of each radius-vector is entered on the 
plates ; and other numerical data and explanations are given. 
Figs. 2 and 3 correspond to /3 = y, and here the figures as computed cross one 
another. The reality must, therefore, be two bulbs joined by a stalk, like a dumb¬ 
bell. The dotted lines have been filled in conjecturally, and must show pretty closely 
what that single figure, formed by the coalescence of the two masses, must be. 
Figs. 4 and 5 show in a similar manner the case of /3 — and here the two masses 
are separate, although nearly in contact. When (3 — \ the shapes present similar 
characters, but are wider apart. 
§ 9, On the Use of Spherical Harmonic Analysis as a Method of Approximation. 
Spherical harmonic analysis gives less accuracy as the bodies considered depart 
more and more from spheres. How far, then, do our results present an approach to 
accuracy ? To answer this question, we have to find how nearly the potentials at the 
surfaces of these figures may be computed from the spherical harmonic formulae. 
It would be laborious to make an accurate computation of the potential, and it 
fortunately appears to be unnecessary to do so, since a sufficient answer may be 
obtained in another way. 
The potential of an ellipsoid of revolution may be computed either rigorously or by 
harmonic analysis. With a certain degree of eccentricity the approximate result will 
agree badly with the rigorous one. 
If the ellipsoid consists of a fluid of unit density, there is a certain angular velocity 
which makes it a level surface. If w be that angular velocity, then we know that 
the spherical harmonic solution would give 1 — 15 <u 2 /167t as the ratio of the minor to 
the major axis. If then c, a, are the rigorous values of the minor and major semi-axes, 
the harmonic approximation'is good if cja does not differ much from 1 — 15ar I677-. 
If we denote by 1 — p the factor by which the approximate value of the ratio of 
the axes is to be multiplied in order to obtain the rigorous value, we have 
_ ,_ c/a ^ 
^ 1 — 15ar/lG7r 
and [i may be regarded as a measure of inaccuracy. 
A table of the values of <o 3 /27t, corresponding to various eccentricities 
c= v /(l — (c/a) 2 ), is computed from the transcendental equation in Thomson and 
Tait’s ‘ Natural Philosophy,’ § 772. From these I compute as follows 
