186 



Mr. G. H. Darwin on 



[Dec. 19, 



but it would be necessary to give a figure, and to go into details, to 

 give the results satisfactorily. 



A similar examination of the equation, giving the retardation of the 

 earth's rotation, shows that there is not so much variety of result, for 

 the tidal friction always tends to retard the earth. 



This completes the consideration of the instantaneous effects on the 

 earth, and the next point demanding attention is the reaction, which 

 the bodily tides have upon the disturbing bodies. 



The problem is solved by the consideration that however the three 

 bodies may interact the resultant moment of momentum of the moon- 

 earth system remains constant, except in so far as it is affected by the 

 sun's action on the earth. The application of this principle results in 

 an equation giving the rate of increase of the square root of the moon's 

 distance in terms of the heights and retardations of the several bodily 

 tides on the earth ; it appears that all the tides, except the fortnightly 

 one, tend to make the moon's distance increase with the time, but the 

 fortnightly tide acts in the opposite direction ; its effect is, however, 

 in general very small compared with that of the other tides. It is 

 proved, also, that the tidal reaction on the sun, which goes to modify 

 the earth's orbit, has quite insignificant effects, and may be neglected. 



I will now show, from geometrical considerations, how some of the 

 results previously stated come to be true. It will not, however, be 

 possible to obtain a quantitative estimate in this way. 



The three following propositions do not properly belong to an 

 abstract, since they are not given in the paper itself; they merely 

 partially replace the analytical method pursued therein. The results 

 of the analysis were so wholly unexpected in their variety, that I have 

 thought it well to show that the more important of them were con- 

 formable to common sense. These general explanations might doubt- 

 less be multiplied by some ingenuity, but it would not have been easy 

 to discover the results, unless the way had been first shown by analysis. 



Prop. I. If the viscosity he small the earth's obliquity increases, the rota- 

 tion is retarded, and the moon's distance and periodic time increase. 

 The figure represents the earth as seen from above the South Pole, 

 so that S is the Pole, and the outer circle the Equator. The earth's 

 rotation is in the direction of the curved arrow at S. The half of the 

 inner circle which is drawn with a full line is a semi- small- circle of S. 

 lat., and the dotted semi-circle is a semi- small- circle in the same N. lat. 



Generally dotted lines indicate parts of the figure which are below 

 the plane of the paper. 



It will make the explanation somewhat simpler, if we suppose the 

 tides to be raised by a moon and anti-moon diametrically opposite to 

 one another. Then let M and M' be the projections of the moon and 

 anti-moon on to the terrestrial sphere. 



