162 PROFESSOR W. THOMSON ON THE SECULAR COOLING OF THE EARTH. 



the expression for v becomes v^-^- V for all positive, and v^—V for all negative 



dv 

 values of ^; and (3), That the expression for y- is the differential co-eflBcient 



with reference to x, of the expression for v. The propositions (1) and (3) are proved 

 directly by differentiation. To prove (2), we have, when ^ = 0, and x positive, 



2V 



-/ dzt 



Vttc 

 or according to the known value, \\/'^, of the definite integral / dzi~^ , 



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 ^ = and x negative, 



t—vi^— V. 



The admirable analysis by which Fouhier arrived at solutions including this, 

 forms a most interesting and important mathematical study. It is to be found 

 in his " Theorie Analytique de la Chaleur." Paris, 1822. 



14. The accompanying diagram represents, by two curves, the preceding 



expressions for -^. 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 uni- 

 form temperature, and suddenly exposed 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 certain average quality as to conductivity and specific heat, 

 the value of f, as I have shown in a previous communication to the Eoyal So- 

 ciety,* will be 400, to unit of length a British foot and unit gf time a year ; and 

 the equation expressing the solution becomes 



dv V 

 dx 35-4 



1 



■ € 



x' 



~ i&oo't . 



5 



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



becomes less than e (which being equal to about 270' niay be regarded as in- 

 sensible), 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 sen- 

 sible at depths exceeding 568 miles, and is therefore confined to so thin a crust, 

 that the influence of curvature may be neglected. 



* Oil the Periodical Variations of Underground Temperature. Trans. Roy. Soc. Edin. 

 March 1860. 



