ON CERTAIN RELATIONS AMONG THE POSSIBLE CHANGES IN THE 



MOTIONS OF MUTUALLY ATTRACTING SPHERES WHEN 



DISTURBED BY TIDAL INTERACTIONS. 



I. INTRODUCTION. 



Sir George Darwin has written a number of classical memoirs on the 

 subject of tidal friction which are remarkable not only for the profundity 

 and the thoroughness of the mathematical analysis, but also for the charm 

 and lucidity of the exposition of the nature of the problems treated, the 

 hypotheses upon which the investigations were based, and the conclusions 

 which were reached. Frequent references will be made to these memoirs 

 in this paper, and for simplicity they will be designated by numbers as 

 follows : 



1. On the bodily tides of viscoxis and semi-elastic spheroids, and on the ocean tides upon a 



yielding nucleus. <Phil. Trans, of the Royal Soo., Part I, 1879, pp. 1-35. 



2. On the precession of a viscous spheroid, and on the remote history of the earth, <Phil. 



Trans., Part II, 1879, pp. 447-538. 



3. On the secular changes in the^ elements of the orbit of a satellite revolving about a 



tidally distorted planet. '<Phil. Trans., Part II, 1880, pp. 713-891. 



4. On the tidal friction of a planet attended by several satellites, and on the evolution of 



the solar system. <Phil. Trans., Part II, 1881, pp. 491-535. 



5. The determination of the secular eflfects of tidal friction by a graphical method. < Pro- 



ceedings of the Royal Society of London, vol. 29 (1879), pp. 168-181. 



Darwin's method of treatment is to express the tide-generating poten- 

 tial as a sum of terms, each of which is the product of a second-order 

 soHd harmonic and a simple time harmonic, and then to derive the corre- 

 sponding surface harmonics which define the tidal deformations when the 

 system has assumed a condition of steady movement. The results are 

 adapted to viscous or elastico-viscous spheroids, the heights and lags of 

 the several tides being expressed in terms of the speeds of the tides and 

 the viscosity, or the rate of decay of elasticity of the tidally distorted body. 

 The effects of these tides upon the motions of the disturbed and disturbing 

 bodies are then derived with rare skill. 



Apparently Darwin's work can be questioned, if at all, only where he 

 applies his analysis to the earth-moon system. Here he reaches the con- 

 clusion that very probably the moon once separated from the earth by 

 fission, and that it has been driven to its present distance by tidal friction. 

 In reading these conclusions we should heed his warning:* 



The result at which I now arrive affords a warning that every conclusion must always 

 be read along with the postulates on which it is based. 



' 2, p. 532, footnote. 



79 



