1886.] 



On a Rotating Mass of Fluid. 



319 



III. " On Jacobi's Figure of Equilibrium for a Rotating Mass of 

 Fluid." By G. H. Darwin, M.A., LL.D., F.R.S., Fellow of 

 Trinity College and Plumian Professor in the University of 

 Cambridge. Received October 12, 1886. 



I am not aware that any numerical values have ever been de- 

 termined for the axes of the ellipsoids, which are figures of equi- 

 librium of a rotating mass of fluid.* 



In the following paper the problem is treated from the point of 

 view necessary for reducing the formulae to a condition for computa- 

 tion, and a table of numerical results is added. 



Let a, b, c be the semi-axes of a homogeneous ellipsoid of unit 

 density ; let the origin be at the centre and the axes of x, y, z be in 

 the directions a, b, c. 



Then if we put — 



^2 =a 2 + ^ £2=&3 + ^ C*=c*+u, and 



f°° du 



(l) 



it is knownf that the potential of the ellipsoid at an internal point 

 x, y, z is given by — 



V=7r bc\* + —— + 1 (2) 



L a da b db c dc J ^ ' 



Now let us introduce a new notation, and let 



* The following list of papers bearing on this subject is principally taken from a 

 report to the British Association, 1882, by W. M. Hicks : — 



Jacobi, ' Acad, des Sciences,' 1834 ; Liouville, ' Journ. Ecole Polytech.,' vol. xiv, 

 p. 289 ; Ivory, ' Phil. Trans.,' 1838, Pt. I, p. 57 ; Pontecoulant, ' Syst. du Monde,' 

 vol. ii. The preceding are proofs of the theorem, and in more detail we have : — 

 C. O. Meyer, 'Crelle,' vol. xxiv, p. 44; Liouville, ' Liouville' s Journ.,' vol. xvi, 

 p. 241; a remarkable paper by Dirichlet and Dedekind, ' Borchardt's Journ.,' 

 vol.lviii, pp. 181 and 217 ; Niemann, ' Abh. K. Gres. "Wiss. Gottingen,' vol. ix, 1860, 

 p. 3 ; Brioschi, ' Borchardt's Journ.,' vol. lix, p. 63 ; Padova, ' Ann. della Sc. 

 Norm. Pisa,' 1868-9 (being Dirichlet and Biemann's work with additions) ; 

 Grreenhill, 4 Proc. Camb. Phil. Soc.,' vol. iii, p. 233 and vol. iv, p. 4 ; Lipschitz, 

 'Borch. Journ.,' vol. lxxviii, p. 245; Hagen, £ Schlomilch Zeitsch. Math.,' vol. xxiv, 

 p. 104 ; Betti, 'Ann. di Matem.' vol. x, p. 173 (1881) ; Thomson and Tait's ' Nat. 

 Phil.' (1883), Part II, §778 ; a very important paper by Poincare, ' Acta Mathem.,' 

 7, 3 and 4 (1885). 



t Thomson' and Tait's 'Nat. Phil.' (1883) §494, I, The form in which the 

 formula is her<} given is slightly different from that in (8), (11), (15) of §§ 494, 7c, I. 



