258 



NATURE 



[Feb. I, 1872 



short of infinite rigidity. This apphcation I have no tiiiie 

 at present to malie ; but without attempting a rigorous 

 investigation, it is easy roughly to estimate an inferior 

 hmit to the admissable rigidity. In the first place, sup- 

 pose, with perfect elasticity, the rigidity be so slight that 

 distorting stress of ,'„ of a gramme weight would produce 

 an angular distortion of a half degree or a degree. 

 The whole would not rotate as a rigid body round 

 one " instantaneous axis " at each instant, but the rotation 

 would take place internally, round axes deviating from 

 the axis of external figure, by angles to be measured in 

 the plane through it and the line perpendicular to the 

 ecliptic in the direction towards the latter line. These 

 angular deviations would be greater and greater the more 

 near we come to the earth's centre, and the greatest 

 angular deviation would be comparable with 1°. Hence 

 the moment of momentum round the solsticial line would 

 be sensibly less than if the whole mass rotated round the 

 axis of figure. Now suppose for a moment our measure- 

 ment of force to be founded on a year as the unit of time. 

 We find the amount of precession in a year by dividing the 

 mean amount of the whole couple due to the influence of 

 moon and sun by the moment of momentum of the earth's 

 motion round the solsticial line. Hence the amount of pre- 

 cession would be sensibly augmented by the elastic yielding ; 

 for the motive couple is uninfluenced by the elastic yielding, 

 if we suppose the earth to be of uniform internal density. 

 An ordinary elastic jelly presents a specimen of the degree 

 of elasticity here supposed, as is easily seen when we con- 

 sider that the mass of a cubic centimetre of such material 

 is a gramme, and therefore that the weight of a cubic 

 centimetre of the substance is the "gramme weight " un- 

 derstood in the specification y'g of a gramme weight per 

 square centimetre. If then, the interior mass of the earth 

 were no more rigid than an ordinary clastic jelly, and if 

 the upper crust were rigid enough to resist any change of 

 figure that could sensibly influence the result, the preces- 

 sion would be considerably more rapid than if the rigidity 

 were infinite throughout. The lunar nineteen-yearly nuta- 

 tion proves a higher degree of elasticity than this ; the solar 

 semi-annual nutation still a higher degree ; and still higher 

 yet the imperceptibility of the lunar fortnightly nutation ; 

 provided always we suppose the interior mass to be of 

 uniform density, and the upper crust rigid enough to per- 

 mit no influential change of figure. 



The motive of the nineteen-yearly precession may 

 be mechanically represented by two circles of matter 

 pivoted on diameters fixed in the plane of the earth's 

 equator, bisecting one another perpendicularly at the 

 earth's centre. These two circles must oscillate round 

 their pivot-diameters, each through an angle of about 

 5° on one side and the other of the plane of the 

 equator, in a period of about nineteen years, to pro- 

 duce the lunar nineteen-yearly nutation (more nearly 

 eighteen years seven months). If the radius of each of 

 the supposed material circles is equal to the moon's mean 

 distance from the earth, the mass of each must be a little 

 less than the moon's mass, and one of them a little less 

 than the other.* The diameter on which the latter is pivot- 

 ed is to be the equinoctial line. The latter alone causes 

 the nutation in right ascension ; the former the nutation 



" The greater is etjual to the moon's mass multiplied by the cosine of the 

 obliquity ol the cchptic ; the less is equal to the moon's mass multiplied by 

 the cosine of twice the obliquity of the ecliptic. 



in declination. The phases of maximum and of zero de- 

 flection, in the oscillations of the two circles, follow alter- 

 nately at equal intervals of tiine, so that when either is in 

 the plane of the earth's equator, the other is at its greatest 

 inclination of 5" on either side. Taking one of the con- 

 stituents of the nutational motive alone, we find, on the 

 principles indicated above, jj^ of a gramme weight 

 per square centimetre as a very rough estimate for the 

 greatest tangential stress produced by it in the material 

 underlying the earth's crust. Now it is clear that the 

 central parts of the earth and the upper crust cannot, in 

 the course of the nutatory oscillations, experience relative 

 angular motions to any extent consid-jrable in comparison 

 with the nutation of the upper crust, without considerably 

 affecting the whole amount of the nutation. The nutation 

 in declination amounts to 9"'25 on each side of the mean 

 position of the earth's poles, and therefore the tangential 

 stress of ^Jy of a gramme weight per square centimetre 

 cannot produce an angular distortion considerable in com- 

 parison with 9". 



An angular distortion of 8" is produced in a cube of 

 glass by a distorting stress of about ten grammes weight 

 per square centimetre. We may, therefore, safely con- 

 clude that the rigidity of the earth's interior substance 

 could not be less than a millionth of the rigidity of glass 

 without very sensibly augmenting the lunar nineteen- 

 yearly nutation. The lunar fortnightly nutation in decli- 

 nation amounts theoretically to about 'i", and it is so small 

 as to have hitherto escaped observation. It probably 

 would have been so large as to have been observed were 

 the interior rigidity of the earth anything less than :^}nao 

 of that of glass, always provided that the upper crust is 

 rigid enough to prevent any change of form sensibly in- 

 fluencing the nutational motive couple. To understand 

 the degree of rigidity meant by " :.^.^„^^-„ of the rigidity of 

 glass," imagine a sheet of some such substance as gutta- 

 percha or vulcanised india-rubber of a square metre area 

 and a centimetre thickness. Let one side of the sheet be 

 cemented to a perfectly hard plane vertical wall, and let 

 a slab of lead 8'8 centimetres thick (weighing therefore a 

 metrical ton) * be cemented to the other side of it. If 

 the rigidity of the substance be .,„,,\,„„ of the rigidity of 

 glass,! ^id 'he range of its elasticity sufficient, the side 

 of the sheet to which the lead is attached will be dragged 

 down relatively to the other through a space of ^.t of a 

 centimetre. 



In the argument from tidal deformations of the solid 

 part of the earth's material, which I communicated to the 

 Royal Society ten years ago, and mentioned incidentally 

 at the recent meeting of the British Association, I 

 showed that though precession and nutation would be aug- 

 mented by want of rigidity in the interior, they would be 

 diminished by want of rigidity in the upper crust, and that 

 on no probable hypothesis can we escape the conclusion 

 that the earth as a whole is less yielding than a globe of 

 glass of the same dimensions and exposed to the same 

 forces. That argument, therefore, proves about 200,000 

 times greater rigidity for the earth as a whole than what I 



^ The metrical ton, or the mass of a cubic metre of water at temperature 

 of maximum density, is "9842 of the British ton The thickness of a slab of 

 lead of a square metre area, weighing a metrical ton, is, of course, equal to 

 a metre divided by the specific gravity of lead. 



t Everett's measurements give 244X10& centimetres waight per square 

 centimetre for the rigidity of the glass on which he experimented. Instead 

 of this I take 240 X lo^, for simplicity. 



