NATURE 



257 



THURSDAY, FEBRUARY i, 1872 



THE INTERNAL FLUIDITY OF THE EARTH 



WE have been favoured with permission to reprint the 

 following extract from a letter addressed by Sir 

 Wm. Thomson to Mr. G. Poulett Scrope: — 



The University, Glasgow, Jan. 15, 1872 

 Dear Sir,— I thank you very much for the copy of 

 your beautiful book on Volcanoes, which you have been 

 so kind as to send me through Professor Geikie. It is 

 full of matter most interesting to me, and I promise 

 myself great pleasure in reading it. 



I see with much satisfaction, in your prefatory remarks, 

 that you " earnestly protest against the assertion of some 

 writers, that the theory of the internal fluidity of the globe 

 is or ought to be generally accepted by geologists on the 

 evidence of its high internal temperature." Your sentence 

 upon the " attractive sensitional idea that a molten in 

 terior to the globe underlies a thin superficial crust ; its 

 surface agitated by tid il waves and flowing freely towards 

 any issue that may here and there be opened for its out- 

 ward escape," in which you say that you " do not think it 

 can be supported by reasoning, based on any ascertained 

 facts or phenomena," is thoroughly in accordance with 

 true dynamics. It will, I trust, have a great effect in 

 showing that volcanic phenomena, far from being de- 

 cisive, as many geologists imagine them to be, in favour of 

 a thin crust enclosing a wholly liquid interior, tend rather, 

 the more thoroughly they are investigated, to an opposite 

 conclusion. 



I must, however, take exception to your next sentence, 

 that in which you say that " M. Delaunay has disposed of 

 the well-known astronomical argument of Mr. Hopkins 

 and Sir W. Thomson, as to the entire or nearly entire 

 solidity of the earth, derived from the nutation of its 

 axis." Delaunay's deservedly high reputation as one of 

 the first physical astronomers of the day, has naturally led 

 many in this country to believe that his objection to the 

 astronomical argument in favour of the earth's rigidity 

 cannot but be valid. It has even been hastily assumed 

 that the objection is founded on mathematical calculation, 

 an error which the most cursory reading of Delaunay's 

 paper corrects. His hypothesis of a viscous fluid breaks 

 down utterly when tested by a simple calculation of the 

 amount of tangential force required to give to any globular 

 portion of the interior mass the precessional and nuta- 

 tional motions, which, with other physical astronomers, he 

 attributes to the earth as a whole. Thus : taking the 

 ratio of polar diameter to equatorial diameter as 299 to 

 300, and the density of the upper crust as half the mean 

 density of the earth, I find (from the ordinary elementary 

 formula;) that when the moon's declination is 28°^, the 

 couple with which she tends to turn the plane of the 

 earth's equator towards the plane of her own centre and 

 the equinoctial line has for its moment a force of '285 X 

 10" times the gravity of one gramme at the earth's sur- 

 face, or rather more than a quarter of a million million tons 

 weight, on an arm equal to the earth's radius. A quadrant 

 of the earth being ten thousand kilometres, the area is 

 V 01. V. 



five hundred and nine million square kilometres, or 5-09 

 million million million square centimetres. Hence a 

 force of '285 X 10"^ grammes weight distributed equally 

 over two-thirds of the earth's area would give '084 of a 

 gramme weight per square centimetre. This supposition 

 is allowable (for reasons with which I need not trouble 

 you) in estimating roughly the greatest amount of tan- 

 gential force acting between the upper crust and a sphe- 

 rical interior mass in contact with it, from the preceding 

 accurate calculation of the whole couple exerted by the 

 moon on the upper crust. It is thus demonstrable that 

 the earth's crust must, as a whole, down to depths of hun- 

 dreds of kilometres, be capable of transmitting tangential 

 stress amounting to nearly yV of a gramme weight per 

 square centimetre. Under any of such stress as this any 

 plastic substance which could commonly be called a 

 viscous fluid would be drawn out of shape with great 

 rapidity. Stokes, who discovered the theory of fluid vis- 

 cosity, and first made accurate investigations of its amount 

 in absolute measure,found that a cubic centimetre of water, 

 if exposed to tangential force of the millionth part of -^ of 

 a gramme weight on each of four sides, would even under 

 so small a distorting stress as this, become distorted so 

 rapidly that at the end of a second of time its four corre- 

 sponding right angles would become one pair of them 

 acute and the other obtuse, by as much as a two-hundredth 

 part of the angle whose arc is radius, that is to say by 

 •29 of a degree. Not as much as a ten-million-millionth 

 part of this distortion could be produced every second of 

 time by the lunar influence in the material underlying the 

 earth's crust without very sensibly affecting precession and 

 nutation ; for the effect of the maximum couple exerted on 

 the upper crust by the moon is to turn the whole earth in a 

 second of time through an angle of a one-hundred-million- 

 millionth of '57 of a degree, so as to give to it at the end 

 of a second the position obtained by geometrically com- 

 pounding this angular displacement with the angular dis- 

 placement due simply to rotation. Hence the viscosity 

 assumed by Delaunay, to produce the effect he attributes 

 to it, must be more than ten million million million times 

 the viscosity of water. How much more may be easily 

 estimated with some degree of precision from Helmholtz's 

 mathematical solution of the problem of finding the 

 motion of a viscous fluid contained in a rigid spherical 

 envelope urged by periodically varying couples.* The 

 most interesting part of the application of this solution to 

 the hypothetical problem regarding the earth, is to find 

 how rapidly the obliquity of the ecliptic would be done 

 away with by any assumed degree of viscosity in the in- 

 terior ; such an amount of viscosity, for example, as 

 would render the excesses of precession and nutation above 

 their values for a perfectly rigid interior, not greater than 

 observation could admit. 



The hypothesis of a continuous internal viscous fluid 

 being disposed of, the question occurs, what rigidity must 

 the interior mass have, even if enclosed in a perfectly 

 rigid crust, to produce the actual phenomena of precession 

 and nutation ? The solutions given by Lamd and myself 

 of the problem of the vibrations of an elastic solid globe, 

 may be readily applied to determine the influences on 

 precession and on the several nutations, which would be 

 produced by elastic yielding with any assumed rigidity 



* Hclmholu and Piotrowski, " Ueber Reibung tropfbarer Fliissigkeiten,* 

 Imp. Acad. Vienna, i860. 



