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IV. On the Figure and Stability of a Liquid Satellite. 
Bxj Sir G. IT. Darwin, K.C.B., F.R.S., Plumian Professor and Fellow of 
Trinity College , Cambridge. 
Received January 7 ,—Read February 8, 1906. 
Table of Contents. 
Page 
Preface.162 
Part I.—Analysis. 
Section 
1. The stability of liquid satellites.163 
2. Figures of equilibrium of a rotating mass of liquid and their stability.167 
3. On the possibility of joining two masses of liquid by a thin neck.170 
4. Notation.176 
5. The determination of gravity on Roche’s ellipsoid.177 
6. Form of the expression for the gravitational lost energy of the system.185 
7. The mutual energy of two ellipsoids.187 
8. Remaining terms in the expression for the lost energy.190 
9. Final expression for the lost energy of the system.192 
10. Determination of the forms of the ellipsoids.193 
11. Solution of the equations.204 
12. Determination of the form of the second ellipsoid.208 
13. Equilibrium of two ellipsoids joined by a weightless pipe.209 
14. Ellipsoidal harmonic deformations of the third order. 213 
15. The values of for higher harmonics.•.218 
16. The fourth zonal harmonic inequality.220 
17. The fifth zonal harmonic inequality.222 
18. Moment of momentum and limiting stability.223 
19. Approximate solution of the problem. 224- 
Part II.—Numerical Solutions. 
20. Roche’s infinitesimal ellipsoidal satellite in limiting stability.228 
21. Roche’s ellipsoidal satellite, of finite mass, in limiting stability, the planet being also 
ellipsoidal.230 
22. Harmonic deformations of the ellipsoids.234 
23. Certain tests and verifications. 235 
24. Figures of equilibrium of two masses in limiting stability.237 
25. Unstable figures of equilibrium of two masses.237 
26. On the possibility of joining the two masses by a weightless pipe. 239 
Part III.—Summary.239 
VOL. CCVI. — A 405. Y 28.6.06 
