TEANSACTIONS OF SECTION A. 483 



but for large ones it is so far from correct that the tendency for the obliquity to 

 vary may become nil ; and for yet larger ones the obliquity may tend to decrease. 



A complete analysis of the state of things for various obliquities and viscosities 

 shows that there is a great variety of positions of dynamical equilibrium, some of 

 which are stable and 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 was shown in the general explanation that the effect on the moon is a force 

 tangential to her orbit accelerating 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 ecliptic, 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 slackening 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 supposi- 

 tion 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 u"'Q 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 

 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 

 acceleration of the moon. He fully admitted that his data did not admit of pre- 

 cise results ; but if I am correct in the present conclusion, 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 viscosity 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 numerically for a degree of viscosity which would 

 make the changes proceed with nearly a maximum rapidity; estimated by the 



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