April 29, 1909] 



NA TURE 



253 



level. If with these simplifying assumptions there 

 is combined the observed fact that the relative change 

 of level is two-thirds of what it would be if the 

 earth were absolutely rigid, it is found, as Lord 

 Kelvin did in effect find, that the calculated rise and 

 fall of the surface is one-third of what it would be 

 if the earth were made of homogeneous incompressible 

 fluid, and the calculated change of its attraction due 

 to the sun, or moon, is one-half of the tide-generating 

 force of the sun, or moon. The rigidity which the 

 material, supposed homogeneous and incompressible, 

 would need to have in order that the two numbers 

 may have the calculated values, \ and \, is about the 

 sarfie as the rigidity of steel. Both the numbers 

 g and 5, which are thus calculated are inferred, partly 

 from a result of observation, and partly from the sub- 

 sidiary hypotheses of homogeneity and incompres- 

 sibility. If these hypotheses are discarded, all that 

 can be inferred from observations of fortnightly tides 

 and horizontal pendulums is a single equation con- 

 necting two numbers. The number which in the 

 special case is 5 is in general conveniently written as 

 ffe, and the number which in the special case is 3 

 may be called fe. The observations in question concur 

 in leading to the equation /j — fe = J. (In the special 

 case l-h = \.) 



It was first suggested by Prof. Simon Newcomb 

 that the length of the earth's free period of nutation, 

 usually called the "Chandler period," may be an 

 independent index of the yielding of the earth to 

 small forces. It has long been known that if the 

 earth were absolutely rigid this period would be about 

 306 days, k free nutation of the earth would be 

 manifested by periodic changes of latitude of places 

 on its surface. Small variations of latitude have long 

 been known to exist, but all efforts of astronomers 

 to detect a period of 306 days in these variations 

 failed. It was announced by Dr. S. C. Chandler, in 

 1S91, that the variations are roughly periodic, but 

 that the period is really 427 days instead of 306. 

 Newcomb pointed out that the lengthening of the 

 period must be due to a yielding of the earth. At 

 any instant the earth is rotating about an axis which 

 does not quite coincide with a principal axis. A solid 

 globe would be deformed by rotation into an oblate 

 spheroid in the same way as a fluid one, but not 

 so much. The inequality of the so-called " centrifugal 

 force," due to the deviation of the instantaneous axis 

 from a principal axis, produces a slight deformation 

 of the surface, accompanied by a slight alteration of 

 the attraction, and these effects can be specified by 

 means of the same two numbers h and fe as are 

 required to express the effects of tidal disturbing 

 forces. Mr. S. S. Hough, H.M.'s Astronomer at the 

 Cape of Good Hope, calculated, in 1896, the lengthen- 

 ing of the period in the case of a solid elastic globe 

 of homogeneous incompressible material. The 

 problem has recently been discussed in a more general 

 way by Prof. G. Herglotz, who was able to dispense 

 with the hypothesis of homogeneity. A review of the 

 theory, as presented by Herglotz', shows that it is 

 possible to dispense with the hypothesis of incom- 

 prepsibility also, and that the lengthening of the 

 period depends upon the number fe, and not upon the 

 number /i. The number fe is found to be expres- 

 sible in terms of the two periods (306 and 427 days), 

 the ellipticity and mean radius of the surface, the 

 angular velocity of rotation, and the mean value of 

 gravity at the surface. This number is therefore 

 known. Its value is found to be about y\. The result 

 that fe = ,-'-- means that the alteration in "the attraction 

 of the earth on account of the distortion produced in 

 it by the sun or moon is actually about four-fifteenths 

 NO. 2061, VOL. 80] 



of the tide-generating force of the sun or moon. This 

 result does not depend upon any hypothesis as to 

 the homogeneity or incompressibility of the material. 

 The only assumptions that are used in obtaining it 

 are the assumption that an equilibrium theory is 

 applicable to the forces in question, and the assump- 

 tion, commonly made in the theory of the figure of 

 the earth since the time of Laplace, viz. that the 

 surfaces of equal density within the earth are main- 

 tained in ellipsoidal shapes by the rotation. The 

 result does not depend upon the special hypothetical 

 law of density which Laplace introduced. Any law 

 of density which satisfies the ordinary laws of 

 hydrostatics will suffice.' 



When the result expressed by the equation fe = T'ir is 

 combined with the result of observations of the tides 

 and horizontal pendulums (h— fe = g), it is found that 

 h = 'l. This result means that the surface of the earth 

 actually yields to the tidal deforming influence of 

 the sun and moon by si.x-twenty-fifths of the amount 

 by which it would yield if the earth were made of 

 homogeneous incompressible fluid. The number -§;, 

 takes the place of Lord Kelvin's number 7V 



The result that the earth actually yields a good 

 deal less than Lord Kelvin supposed it to do suggests 

 that it is decidedly more rigid than he estimated it 

 to be. There are, however, many difficulties in the 

 way of a more precise estimate, the chief being the 

 heterogeneity of the material. If this fact is dis- 

 regarded, and the simplifying assumption of homo- 

 geneity is made, it appears to be impossible to satisfy 

 both the equations h = ^i and fe=i*.-. An additional 

 difficulty arises from the compressibility of the 

 material, but, although this cannot be met directly, it 

 is not very serious, because the general effect of 

 compressibility must almost certainly be that any esti- 

 mate of rigidity based on the simplifying assumption 

 of incompressibility is under the mark. A possible 

 method of procedure is to assume the earth to consist 

 of a central nucleus of incompressible material of one 

 density and rigidity, enclosed in a shell oT incompres- 

 sible material of a smaller density and a different 

 rigidity, in the manner advocated by Prof. 

 E. Wiechert, who regards the earth as made up of 

 an iron core enclosed in a rocky shell. This method 

 was developed by Dr. W. Schwe'ydar, who found that, 

 with the densities proposed by Wiechert, the rigidity 

 of the core would have to be nearly three times that 

 of steel, and the rigidity of the shell about one-eighth 

 of that of steel. The possibility of a comparatively 

 small rigidity in the enclosing shell suggests that 

 there may be within it, or between it and the core, 

 a layer of molten rock, devoid of rigidity, such as has 

 sometimes been invoked in connection with the 

 explanationof seismic and volcanic phenomena. This 

 hypothesis is found, when tested mathematically, to 

 require much too great rigidities both of the core and 

 of the outer part of the shell. It appears, however, 

 to be quite possible that the earth may consist of a 

 very dense and very rigid core enclosed in, and con- 

 nected by solid matter with, a lighter shell or crust, 

 the greater part of which is solid and of a rigidity 

 comparable with that of granite (about one-third of 

 that of steel), the shell being honeycombed with 

 hollow spaces containing molten matter. But it seems 

 to be impossible that the molten matter should form a 

 continuous layer separating the outer portions of the 

 earth's body from the inner portions. 



1 Since the paper was written and sent in to the Royal Society. Prof. Lar- 

 mor has shown that the result is independent of the supposed ellipsoidal 

 shape of the surfaces of equal density. It is therefore established, quire 

 generally, for any constitution of the earth which would admit of the 

 applica'ion of an equilibrium theory to forces of the type in question. It 

 is practically certain that the actual constitution is such that a theory of 

 this kind can be applied. 



