7L4 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Page 
§ 20. Secular changes in the proper planes of the earth and moon when the 
viscosity is large . 825 
§ 21. Graphical illustration of the preceding integrations...... 831 
§ 22. The effects of solar tidal friction on the primitive condition of the earth 
and moon... 834 
V. Secular Changes in the Eccentricity of the Orbit. 
§ 23. Formation of the disturbing function... 836 
§ 24. Secular changes in eccentricity and mean distance .. .. 849 
§ 25. Application to the case where the planet is viscous... 851 
§ 26. Secular change in the obliquity and diurnal rotation of the planet, when the 
satellite moves in an eccentric orbit. 856 
§ 27. Verification of analysis, and effect of evectional tides. 858 
VI. Integration for Changes in the Eccentricity of the Orbit. 
§ 28. Integration in the case of small viscosity... 860 
§ 29. The change of eccentricity when the viscosity is large .„. 864 
VII. Summary and Discussion of Results. 
§ 30. Explanation of problem.—Summary of Parts I. and II. 864 
§ 31. Summary of Part III. 867 
§ 32. Summary of Part IV. 871 
§ 33. On the initial condition of the earth and moon . 876 
§ 34. Summary of Parts V. and VI. 877 
VIII. Review of the Tidal Theory of Evolution as applied to the Earth and the other 
Members of the Solar System. 879 
Appendix.—A graphical illustration of the effects of tidal friction when the orbit of the 
satellite is eccentric . 886 
Introduction. 
The following paper treats of the effects of frictional tides in a planet on the orbit of 
its satellite. It is the sequel to three previous papers on a similar subject.* 
The investigation has proved to be one of unexpected complexity, and this must be 
my apology for the great length of the present paper. This was in part due to the 
fact that it was found impossible to consider adequately the changes in the orbit of the 
satellite, without a reconsideration of the parallel changes in the planet. Thus some 
of the ground covered in the previous paper on “ Precession ” had to be retraversed; 
but as the methods here employed are quite different from those used before, this 
repetition has not been without some advantage. 
* “ On the Bodily Tides of Viscous and Semi-elastic Spheroids, and on the Ocean Tides upon a Yielding 
Nucleus,” Phil. Trans., Part I., 1879. 
“On the Precession of a Viscous Spheroid, and on the remote History of the Earth,” Phil. Trans., 
Part II., 1879. 
“ On Problems connected with the Tides of a Viscous Spheroid,” Phil. Trans., Part II., 1879. 
These papers are hereafter referred to as “ Tides,” “ Precession,” and “ Problems ” respectively. 
There is also a fourth paper, treating the subject from a different point of view, viz. : “ The Determi¬ 
nation of the Secular Effects of Tidal Friction by a Graphical Method,” Proc. Roy. Soc., No. 197, 1879. 
And lastly a fifth paper of more recent date, “ On the Analytical Expressions which give the History of a 
Fluid Planet of Small Viscosity, attended by a Single Satellite,” Proc. Roy. Soc., No. 202, 1880. 
