224 



NATURE 



[January 3, 1895 



had any reason to suppose such dislributions might have 

 occurred in the case in dispute. This he has failed to do— he 

 has evaded the point. (3) Prof. Pearson descends to vague 

 generalitie-i except in regard to Dr. Oliver Lodge, who may be 

 left to dclend himself. 



With the last paragraph of the letter, however, I heartily 

 concur. There is nothing the S.P.R. would welcome more 

 than intelligent and independent criticism. Only the critic would 

 have to study the evidence first, and the Professor apparemly 

 has the '• scientific acumen " to see that by doing so he would 

 cut his own throat ; for he would, ipso facto, become a psychical 

 researcher ! Edward T. Dixon. 



Cambridge, December 29, 1S94. 



ON THE AGE OF THE EARTH. 



IT has been thought advisable to publish the following 

 documents. On October 12 I put my views before 

 Prof. Fitzgerald and Dr. Larmor. The first paper is a 

 copy of my letter to Dr. Larmor. It has now been edited 

 a little, as originally it was rather hurriedly written. 

 Some long mathematical notes, added on .November i, 

 to prove the legitimacy of my appro.ximate method 

 of calculation, are now omitted, as Mr. Heaviside has 

 given exact solutions, and has found that there is 

 practically no difference between mine and the exact 

 numerical answers. That Mr. Heaviside should have 

 been able, in his letters to me during eleven days, to work 

 out so many problems, all seemingly beyond the highest 

 mathematical analysis, is surely a triumph for his new 

 methods of working. Only for Prof. Fitzgerald's encour- 

 agement and sympathy, it is very probable that this 

 document would never have been published. 



I have sometimes been asked by friends interested 

 in geology to criticise Lord Kelvin's calculation of the 

 probable age of the earth. 1 have usually said that it is 

 hopeless to expect that Lord Kelvin should have made 

 an error in calculation, liesides, in every class in mathe- 

 matical physics in the whole world since 1862 the problem 

 has been put before students, and, as the subject is of 

 enormous interest, if there had been any error it cer- 

 tainly would have been discovered before now. 



I dislike very much to consider any quantitative 

 problem set by a geologist. In nearly every case the 

 conditions given are much too vague for the matter to be 

 in any sense satisfactory, and a geologist does not seem 

 to mind a few millions of years in matters relating to 

 time. Therefore I never till about three weeks ago 

 seriously considered the problem of the cooling of the 

 earth except as a mere mathematical problem, as to 

 which definite conditions were given. But the best 

 authorities in geology and paUtontology are satisfied 

 withevldences in their scicncesof a much greater age than 

 the one hundred million years stated by Lord Kelviri ; 

 and if they are right, there must be something wrong in 

 Lord Kelvin's conditions. On the other hand, his cal- 

 culation is just now being used to discredit the direct 

 evidence ol geologists and biologists, and it is on this 

 account that I have considered it my duty to question 

 Lord Kelvin's conditions. 



The original object of Lord Kelvin's investigation is 

 usually forgotten. He sought to prove, and proved, that 

 the earth is losing energy at a calculable rate. He said 

 that the loss might be the loss of potential or chemical 

 energy instead of sensible heat, or as well as heat, 

 although he thought that a large proportion of 

 potential or chemical eneigy was improbable ; and it is 

 only on the assumption that the earth is a cooling body 

 losing energy originally only of the sensible-heat form, 

 that bis calculation of the age of the earth is based. 

 Not only so, but also his earth is a homogeneous mass of 

 rock such as we have on the surface, with the same con- 

 ductivity and other heat properties. He starts with the 



NO. 1314, VOL. 5'] 



knowledge that there is an average increase of tempera- 

 ture downwards in the earth of one Fahrenheit degree 

 for every 50 feet. Assuming that the earth, a solid, was 

 once at the uniform temperature of 7000 F., that 

 its surface was suddenly brought to and kept at the 

 temperature o, and taking /vt" ik being conductivity 

 and c capacity for heat of unit volume, in year foot units) 

 as 400, he finds that lo"" years have sufficed to cause the 

 temperature-gradient at the surface to be what it is now. 

 He stated that the conditions were sufliciently repre- 

 sented by an infinite uniform mass of matter at 7000' 

 F. with an infinite plane face kept at o. 



At first I preferred to consider a globe of 4.000 miles 

 radius of constant surlace-emissivity to be cooling as if 

 in an enclosure, kept at constant temperature. 1 nude 

 the emissivity infinite, and obtained Lord Kelvin's 

 answer for temperature-gradient near the surface. When 

 the emissivity is taken of a finite value, the time taken 

 to produce the present temperature-gradient is less than 

 Lord Kelvin's answer. 



It is interesting to notice that if we take our enclosure 

 to be at a zero of temperature which we can choose as 

 we please, we have a method of using Fourier's expres- 

 sion in certain cases in which the emissivity is not con- 

 stant. I3y no method of working does it seem probable 

 that we shall greatly alter Lord Kelvin's answer. 



Modification of Lord Kelvin's Conditions. 



But, when we depart from homogeneity, when we 

 assume that the interior of the earth may be of better 

 conducting material than the surface rock in which the 

 temperature-gradient is alone measured, we find a very 

 different state of things from that considered by Lord 

 Kelvin. The cooling from a constant temperature of an 

 infinite mass bounded by a cold plane face, a slice of 

 which near the surface is of inaterial different from the 

 rest of the infinite block, is a problem difficult to attack 

 mathematically. But if the slice is thin, or if much 

 time has elapsed, the following artifice leads to a 

 solution. 



Imagine an infinite homogeneous block, originally at 

 temperature V,, whose surface is kept at o. If ii is 

 sufficiently small and / great, we may neglect the ex- 

 ponential term, and (:' being temperature and /time, and 

 .r ' " 



• the distance from the cold face) 



Vi -i- sJtkJ; 7'i at .ii = V,j-, 



dx 



^/wKj/. 



Rate of liow of heat across unit area at .1 , = i\V, -7- 

 s'ttk,/. I take i as conductivity, and k as conductivity 

 divided by capacity for heat of unit volume. 



Now take another such homogeneous infinite block of 

 different material, and use the letters with affix 2 instead 

 of I. Let the time be the same in both. Let the sur- 

 face slice from 1, to o in the first, and from .r. to o in the 

 second be considered. We can, by taking proper values 

 of V| and V. and .r, and .v-., make the rates of How of 

 heat equal and the temperatures equal at .r, and .v., : 



/•,V,/ Vi = ^••jVj/Vj and V,..,/v'ki = V,.s\Vj 



Hence -(•, -H r, = i:, -i- x... Thus if /;/-.. = X„ we take 



nX; = X,. r , I 



Now we can take the slice x., to o from the second 

 block and let it take the place of the slice x, to o on the 

 first block. The artificial block so produced will go on 

 cooling, its outside face being kept at o,. But we shall 

 have at the point of junction a sudden nniltiplication of 

 dv dr. In fact, dr dx will be what it used to be towards 

 the interior, but will be // limes as great towards the 

 surface. It is of no consequence what the value of «.. 

 is, if times are great and slices thin, the only impurlant 

 thing is that /•, shall be n times k,. The application ol 

 the result is obvious : — 



