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XIII. On Figures of Equilibrium of Rotating Masses of Fluid. 
By G. H. Darwin, M.A., LL.D., F.R.S., Felloio of Trinity College , and Plumian 
Professor in the University of Cambridge. 
Received April- 28,—Read June 16, 1887. 
[Plates 22, 23.] 
In a previous paper* I remarked that there might be reason to suppose that the 
earliest form of a satellite might not be annular. Whether or not the present inves¬ 
tigation does actually help us to understand the working of the nebular hypothesis, 
the idea there alluded to was the existence of a dumb-bell shaped figure of equili¬ 
brium, such as is shown in the figures at the end of this paper. These figures were 
already drawn when a paper by M. Poincare appeared, in which, amongst other 
things, a similar conclusion was arrived at. My paper was accordingly kept back in 
order that an attempt might be made to apply the important principles enounced by 
him to this mode of treatment of the problem. The results of that attempt are, for 
reasons explained below, given in the Appendix. 
The subject of figures of equilibrium of rotating masses of fluid is here considered 
from a point of view so wholly different from that of M. Poincare that, notwith¬ 
standing his priority and the greater completeness of his work, it still appears worth 
while to present this paper. 
The method of treatment here employed is simple of conception; but it is unfor¬ 
tunate that, to carry out the idea, a very formidable array of analysis is necessary. 
In the last section a summary will be found of the principal conclusions, in which 
analysis is avoided. 
§ 1. Formulas of Spherical Harmonic Analysis. 
Let there be two sets of rectangular axes, as shown in fig. 1; and let 2 be 
measured from o to O, whilst Z is measured from 0 to o ; let r 2 = x 2 -f- y 2 -f- 2 2 , 
R 2 = X 2 + Y 2 + Z 2 ; and let c = oO. 
Then 
x -f X = 0, y-f Y = 0, zZ — c .(1) 
Let Wi, Wi, denote the solid zonal harmonics of degree i of the coordinates x, y, z, 
and X, Y, Z, respectively. 
Now we shall require to express the solid zonal and certain tesseral harmonics of 
* ‘ Phil. Trans.,’ Part II., 1881, p. 534. 
3 c 2 
■ 21 . 11 . 87 . 
