January 3, 1895J 



NATURE 



225 



Let the interior of the earth be a uniform sphere, 

 uniformly heated to 7000' F. Take its icas tii times what 

 Lord Kelvin took it, then an increase of temperature 

 downwards from the surface of I F. degree for every 

 50 n feet would be produced in loV/- in years. Take its 

 k as « times what Lord Kelvin takes. Now if we imagine 

 a skin removed and replaced by one of i «th of the 

 thickness and i n\.\\ of the conductivity, that is, take it of 

 Lord Kelvin's conductivity of rock, the surface slope will 

 be I in 50, what it is now, and Lord Kelvin's time will 

 be increased in the proportion ri-'in. 



Considering the great differences in conductivity of 

 such bodies as we know, it is quite conceivable in our 

 knowledge and ignorance of the interior of the earth that 

 «- III may be considerable even now, and probably was 

 very considerable in past times. Roughly we may say 

 that Lord Kelvin's age of the earth, 10'* years, ought to 

 be multiplied by two to six times the ratio of the internal 

 conductivity to the conductivity of the skin. 



I am not in a position to criticise the arguments from 

 tide phenomena which Lord Kelvin or Mr. Darwin would 

 now put forward on the subject of much internal fluidity 

 of the earth. The argument from precession has been 

 given up. Of course much internal fluidity would prac- 

 tically mean infinite conductivity for our purpose. But 

 there is no doubt of a certain amount of fluidity inside 

 even now, and taking it that the inside of the earth is a 

 honeycomb mass of great rigidity, partly solid and partly 

 fluid, we have reason to believe in very much greater 

 quasi-conductivity inside than of true conductivity in the 

 surface rocks, and if there is even only ten times the 

 conductivity inside, it would practically mean that Lord 

 Kelvin's age of the earth must be multiplied by 56.' 



If we imagine the earth perfectly conducting inside 

 with a thin covering, say 60 miles thick, of rock, such as 

 we know it on the surface, we must leave Lord Kelvin's 

 infinite mass and study the sphere. Indeed, if we take 

 it that we have now an infinite mass at 7000 F. of infinite 

 conductivity, cooling through rock of from 60 to 70 miles 

 thick wah a constant gradient of I for every 50 feet, we 

 can imagine that this state of things has existed for an 

 infinite time, and any original distribution of tempera- 

 ture in the rock would settle down to such a state. 



Taking, then, an internal sphere of infinite conduc- 

 tivity - (and working in C.G.S. Centigrade units), its 

 specific heat 02, and the conductivity of the rock o'O02, 

 I find that if at the beginning of time there was an 

 increase of 1 Centigrade in 45 feet, and now there is an 

 increase of i Centigrade degree in 90 feet, the lapse of 

 time is 28,930 million years, or 290 times Lord Kelvin's 

 age, and the core has cooled from Sooo 10 4000 degrees. 

 Or, again, in the last 10'* years the gradient has only 

 diminished by i 400th of its present value, and the core 

 has only changed from 4010 to 4000 degrees. 



I do not know that this speculation is worth much, 

 except to illustrate in another way the augmented answer 

 when we have higher conductivity inside. It would evi- 

 dently lengthen the time if I assumed that the tempera- 

 ture-gradient was not uniform in the shell, but the exact 

 mathematical calculation is so troublesome that I have 

 not attempted it ' John Pkrrv. 



31 Brunswick Square, London, W.C, October 14. 



' Observe th.it, even if we assume th.it there is the same conductivity in- 

 side and outside, inasmuch as the density is Rrcaler, c is greater, say 3 

 tiijics as great, and even without the assistance ot increased conductivity 

 mside. we have 3 times Lord Kelvin's age. I .admit that all such spectilation 

 .-vs to the value uf tis too vague to be of much importance. 



- If 0^, and were the internal temperatures at the times i^ and /, if b is 

 thethicl^ncss of the crust and R ihj radiusof the internal sphere, if s is its 

 speciBc heat and p its density and k its conductivity, 



t-t^ 



_K6sp 

 ' 3* 



log "^K 



3 If '006 be taken as the conductivity of rock, the times are only a third 

 of what 1 have given. 



In connection with this matter I notice that in Lord Kelvin's very short 

 p.il>er, entitled ''Ihe 'Doctrine of Uniformity" in Geology briefly 



NO. I 3 14, VOL. 5 l] 



October 22, 1894. 



The reasoning in my paper was applied either to 

 infinite blocks of cooling material or to a sphere with an 

 internal core which has infinite conductivity. At the 

 time of writing I did not see my way to the consideration 

 of a sphere with a core of finite conductivity and a shell 

 of rock as a covering, but the case is really easy to work 

 when the shell is only a few miles in thickness, as will be 

 seen below. 



Prorlem.— A sphere of radius R = 638 x lo'^centim. 

 of conductivity k = o'47 (or 79 times that of surface-rock) 

 and kic = 0-16464 (or 14 times that of surface-rock), has 

 upon it a shell of rock of thickness 4 x lo'centim. (about 

 2.', miles). The whole mass was once at a temperature 

 V' = 4000' C, and suddenly the outside of the shell was 

 put to o" C. and kept at that. Find the time of cooling 

 until the temperature-gradient in the shell has become 

 I Centigrade degree in 2743 centim. (or 1° F. in 50 

 feet). 



Now, if we are allowed to assume that the shell very 

 rapidly acquired and retained a uniform temperature- 

 gradient throughout its thickness, and it is easy to show 

 that this assumption is allowable (or if not, then the 

 discrepance is in favour of a greater age for the earth), 

 the problem is exactly the same as this : — The above- 

 mentioned sphere has no shell of rock round it, but 

 emits heat to an enclosure of o' C, the constant emissi- 

 vity of its surface' being E = 1-475 X lo"" ; find the 

 time in which the surface-temperature f' becomes 146' C. 



This problem is solved by Fourier, who gives for the 

 temperature at the distance r from the centre 



_ 2VER ,sin erIK r^''""^' 



k er/[<. e cosec e ~ cos e 



where in the successive terms the values of e to be taken 

 are the successive roots of the equation 



(f/tan e = I - ER/>C-. 

 In the present case ER /■ — 20, and r,, e.^ t',, &c., are 

 nearly t, 27r, 37r, ^c. I have, however, taken the actual 

 values oie^ and e.,- two exponential terms, only, being of 

 importance, and I findthat, if / = 96 X lo** years, 

 v' - 1427 + 5*65 = 148-4; 



ist term 2nd icrm 



so that the age of cooling to the present temperature- 

 gradient is more than 96 X 10^ years. 



Refuted," read before the Royal Society of Edinburgh in 1065, he finds : — 

 '■ But the heat which we know, by observation, 10 be now conducted out of 

 the Earth yearly is so great that if this action had been going on with any 

 approach to uniformity for 30,000 million years, the anmunc of heat lost out 

 ot the Earth would have been about as much as would heat, by 100" Cent., 

 a quantity o( ordinary surface-r(^ck 100 limes the Earth's bulk." (The italics 

 are mine.) In his address on "Geological Dynamics," Part II., published in 

 1869 (p. i'j6, vol. ii. " Popular Lectures and Addre.'ises"). he calculates the 

 total amount of energy which wr/j* once have been possessed by the Eartli 

 mass, partly gravitational and partly chemical, as " being about 700 limes as 

 much heat as would raise the temperature of an equal mass of surface-rock 

 from 0° to ico' Cent." (The italics are mine.) I do not think that these two 

 siatenienis hiive ever before been put in juxtaposition. Comparing them, 

 we may say that, according to Lord Kelvin's own figures, if the present 

 action had been going on with any approach to uniformity for lo''" years the 

 amount of heat lost by the Earth w.aild have Leen the i/ySooth part of 

 the whole energy whicli the whole Earth may once have possessed, or 

 i/223oth part ol what Lord Kelvin gives as an estiniaie, an over-estimate he 

 calls it (l>ut he says that it is not possible to make one much less vague), of 

 the whole amount of heat at present in the Earth. I mention this because 

 some mathematical physicists beheve that Lord Kelvin baited his age of the 

 Earth upon a calculation of this total lo^s. He only used it in oppo-^ition to 

 the extreme docirme o' uniformiiy for the past 20,000 million years (a doc- 

 trine which is not now believed in by any geologist), but it lends no sup- 

 port to his calculated age ol the Earth;. 



All through this paper I give ro^ years as Lord Kelvin's age of the Earth. 

 His own words {Trans. /\.S. Edin.y 1862(7) are: — "We must, therefore, 

 allow Very wide limits in such an estimate as I have attempted to make; 

 but [ think we may with much probability say th.at the consoli^fation cannot 

 have taken place less than 20,000,000 years ago, or we should have more 

 underground heat than we actually have [he means a more rapid increase of 

 temperature downwards], nor more than 400.000.000 years ago, or we should 

 not have so much as the least observed underground increment t f tempera- 

 ture." Taking the average difTusivity for heat of the Edinburgh experi- 

 ments, he finds (r) that the present temperature-gradient of 1 Fahr. degree 

 for every 50 feet gives a life of lo^' years. 



■' Because if ;»' is the surface-temperature of the sphere and b the thick- 

 ness of the shell of rock, f'/iS was the surface-gradient in the shell and t-'/^ 

 cnultiplied by conductivity of rock is eiiual to Ef'. 



