Prof. W. Thomson on the Secular Cooling oj the Earth. 7 



and for all values of t, the second term has equal positive and 

 negative values for equal positive and negative values of x, so 

 that when t=0 and x is negative, 

 v = v — V. 



The admirable analysis by which Fourier arrived at solutions 

 including this, forms a most interesting and important mathe- 

 matical study. It is to be found in his Theorie Analytique de la 

 Chaleur (Paris, 1822). 



14. The accompanying Plate (PI. I.) represents, by two curves, 



the preceding expressions for -y- and v respectively. 



15. The solution thus expressed and illustrated applies, for a 

 certain time, without sensible error, to the case of a solid sphere 

 primitively heated to a uniform temperature, and suddenly ex- 

 posed to any superficial action, which for ever after keeps the 

 surface at some other constant temperature. If, for instance, 

 the case considered is that of a globe 8000 miles diameter of 

 solid rock, the solution will apply with scarcely sensible error for 

 more than 1000 millions of years. For if the rock be of a cer- 

 tain average quality as to conductivity and specific heat, the value 

 of k, as I have shown in a previous communication to the Royal 

 Society*, will be 400, to unit of length of a British foot and 

 unit of time a year ; and the equation expressing the solution 

 becomes 



dx 35-4 * 



and if we give t the value 1,000,000,000, or anything less, the 

 exponential factor becomes less than e -5 ' 6 (which being equal to 

 about 2^-q, may be regarded as insensible) when x exceeds 

 3,000,000 feet or 568 miles. That is to say, during the first 

 1000 million years the variation of temperature does not become 

 sensible at depths exceeding 568 miles, and is therefore confined 

 to so thin a crust that the influence of curvature may be neglected. 



16. If, now, we suppose the time to be 100 million years from 

 the commencement of the variation, the equation becomes 



dv V ; f!_ 



-rr- = A ~ ,~~~ € 160000000000 . 



dx 354000 



The diagram (PL I.) therefore shows the variation of temperature 

 which would now exist in the earth if, its whole mass being first 

 solid and at one temperature 100 million years ago, the tempe- 

 rature of its surface had been everywhere suddenly lowered by 



* "On the Periodical Variations of Underground Temperature/' Trans. 

 Roy. Soc. Edinb. March 1860. 



