Sept, 26, 1878] 



NATURE 



A complete analysis of this state of things for various obli- 

 quities and viscosities shows that there is a great variety of 

 positions of dynamical equilibrium, some of which are stable 

 ^nd some unstable. 



Although there is all this variety with respect to the change of 

 the obliquity, yet the tidal friction always tends one way, 

 namely, to stop the earth's rotation. 



It has already been remarked in the general explanation that 

 the effect on the moon is a force tangential to her orbit accele- 

 rating her linear motion, and thus indirectly retarding her 

 angular motion. But it appears that for a very great degree of 

 stiffness and for large inclinations of the earth's axis to the 

 e'cliptic, this force on the moon may be actually reversed ; so 

 that the retardation of the moon's motion may actually be 

 replaced by an acceleration. 



To a terrestrial observer, however, unconscious of the slacken- 

 ing of the earth's diurnal rotation, it would be indifferent whether 

 the moon were undergoing true retardation or true acceleration, 

 for in every case there would result an apparent acceleration of 

 the moon's mean motion. 



It is obvious from what has been said that we have the means 

 of connecting the heights and lagging of the bodily tides in the 

 earth with an apparent secular acceleration of the moon's mean 

 motion. I have applied these ideas to the supposition that the 

 moon has an apparent secular acceleration of 4" per century, and 

 I find that if the earth were a homogeneous viscous spheroid, 

 then the moon must be undergoing a secular retardation of 3"'6 

 per century, while the earth (considered as a clock) must be losing 

 14 seconds in the same time. Under these circumstances the 

 effective rigidity of the earth must be so great that the bodily 

 diurnal and semi-diurnal tides would be quite insensible; the 

 bodily fortnightly tide would, however, be so considerable that 

 the oceanic fortnightly tide would be reduced to one-seventh of 

 its theoretical amount on a rigid nucleus, and the time of high 

 water would be accelerated by three days. 



The supposition that the earth is a nearly perfectly elastic 

 body leads to very different results, which, however, I must now 

 pass over. 



From this and various other considerations, I arrive at the 

 conclusion that the earth has a very great effective rigidity, and 

 that the apparent acceleration of the moon's motion affords no 

 datum for determining the amount of tidal friction on the earth. 



Sir William Thomson has made some interesting remarks 

 about the probable age of the earth in connection with tidal 

 friction, and he derived his estimate of the rate at which the 

 diurnal rotation is slackening principally from the secular accele- 

 ration of the moon. He fully admitted that his data did not 

 admit of precise results, but if I am correct in the present con- 

 clusion, it certainly appears that his argument must lose part of 

 its force. 



The investigation of the secular changes, which such a system 

 would undergo, is surrounded by great mathematical difficulties, 

 but I think that I have succeeded in surmounting them by 

 methods partly analytical and partly arithmetical. 



In a communication of the present kind it would be out of 

 place to consider the methods employed, and I will therefore 

 only speak of some of the results. 



There are two standards by which we may judge of the vis- 

 cosity in the present problem — first the ordinary one, in which 

 it is asserted that it requires so many pounds of tangential stress 

 to the square inch to shear an inch cube through so much in 

 such and such a time ; and secondly, when the viscosity is judged 

 of by the amount by which the behaviour of the spheroid departs 

 from that of a perfectly fluid one ; a numerical value for this 

 sort of measure is afforded by the angle by which the crest of 

 the tidal spheroid precedes the moon, when the obliquity to 

 the ecliptic is zero. 



Now it appears that if the earth possessed a viscosity which 

 was not at all great as estimated by the tidal standard, yet the 

 materials of the earth, when considered in comparison with the 

 substances which we know, would be found to be a substance of 

 very great stiffness — stiffer than lead, and perhaps nearly as stiff 

 as iron. I see, therefore, no adequate reason why some part of 

 the changes, which will be considered presently, should not have 

 taken place during geological history. 



The problem was solved niimerically for a degree of viscosity, 

 which would make the changes proceed with nearly a maximum 

 rapidity. Estimated by the tidal standard, this is neither a very 

 great nor a very small viscosity, for the crest of the semi-diurnal 

 tide precedes the moon by 17* 30'. 



I found, then, that if the changes in the system are tracked 

 back for fifty -six million years, we find the day reduced to six 

 hours fifty minutes, the obliquity to the ecliptic 9° less than at 

 present, and the moon's period round the earth reduced to one 

 day fourteen hours. 



This very short period for the moon indicates of course that 

 her distance from the earth is small. As the moon goes on 

 approaching the earth the problem becomes much more com- 

 plex, and, for periods more remote than fifty-six million years 

 ago, I abandoned the attempt to obtain a scale of times. The 

 solution up to this point shows that the times requisite for these 

 causes to produce such startling effects are well within the time 

 which physicists have admitted to have elapsed since the earth 

 existed. 



From this point in the solution the parallel changes of the 

 obliquity, day and month, were traced without reference to time. 



It appears, then (still looking backwards in time), that the 

 obliquity will only continue to diminish a little more beyond the 

 point already reached ; for, when the sidereal month has be- 

 come equal to twice the day, there is no longer any tendency for 

 the obliquity to diminish, and for yet smaller_values of the month 

 the tendency is to increase again. 



From this we learn that, when the day is equal to or greater than 

 half the month, the position of the earth's axis at right angles 

 to the plane of the moon's orbit is one of dynamical stability. 

 The whole decrease of obliquity from the present value back to 

 the critical point, where the month is equal to twice the day, is 

 10°. From this point in the solution back to the initial state to 

 which the earth and moon are tending, the obliquity to the plane 

 of the lunar orbit was neglected. I then found that the limiting 

 condition, beyond which it was impossible to go, was one in 

 which the earth and moon are rotating, fixed together as a rigid 

 body, in five hours and forty minutes. This condition was also 

 found to be one of dynamical instability, so that, if the month 

 had been a little shorter than the day, the moon must have 

 fallen into the earth, but if the month had been a little longer 

 than the day the moon must have receded from the earth, and 

 have gone through the series of changes, which were traced back- 

 wards up to this initial condition. 



This periodic time of the moon of five hours forty minutes 

 corresponds to an interval of only 6,000 miles between the 

 moon's centre and the earth's siurface. Moreover, if the earth 

 had been treated as heterogeneous instead of homogeneous, this 

 interval between the primeval earth and moon would have been 

 yet further diminished, as also would be the common periodic 

 time. 



The conclusion, therefore, appears to me almost irresistible 

 that if the moon and earth were ever molten viscous bodies, then 

 they once formed parts of a common mass. 



With respect to the obliquity of the ecliptic, the question is 

 one of considerable difficulty, but, on the whole, I incline to the 

 view that, while a large part of the obliquity may be probably 

 referred to these causes, yet that there remains an outstanding 

 part which is not so explicable. 



Besides the results, of which the outlines have been given, 

 I have obtained some others which, as I believe, will aid in the 

 formation of a modified edition of the nebular hypothesis — such 

 as some of the changes to which an annular satellite would be 

 subjected. 



One of the collateral results, which appeared in considering 

 the secular changes of such a system as the earth, moon, and 

 svm, was that a large amount of heat would have been generated 

 in the interior of the earth by means of friction. If, then, it is 

 permissible to suppose that any considerable part of these changes 

 had taken place during geological history. Sir William Thom 

 son's problem of the secidar cooling of the earth would require 

 some modification. 



The magnitude of the undertaking has not allowed me time 

 as yet to apply these ideas to the questions of the eccentricity 

 and inclination of the orbit of the satellite, nor to the cases of 

 other planets besides the earth. 



I think, however, that I see in Asaph Hall's wonderful dis- 

 covery of the Martian satellites, a confirmation of this theory. 

 Their extreme minuteness has, I think, preserved them as a 

 standing memorial of the primitive period of rotation of that 

 planet. The Uranian system, on the other hand, appears, at 

 least at first sight, a stumbling-block. 



It is easy to discern in the planetary system many vera causes 

 which tend to change its configuration, but it is in general very 

 hard to give any quantitative estimate of their effects. 



