I907.] AND CONTRACTION OF THE EARTH. 227 



Ti, T2, T3 . . . Ti start from a common epoch / = o, and the functions 

 V, V\ V" , . . . F* are all equal, the subscripts in the right member of 

 (32) drop out, and the equation reduces to 



e. = (H) + (O — 0i). 



But in the more general problem the functions t^, t^, T3, . . . n date 

 from i chosen epochs, and this gives the means of approximating 

 any curve without change of flexure between x = o, and x == x. 

 § 15. Fourier^ s Methods Adequate for Effecting a Rigorous Solu- 

 tion of the Problem when the Conditions are Known. 

 As already remarked the earth is so large and the crust so thin 

 compared to the length of the radius, that the curvature of the sur- 

 face may be neglected ; and the layer of rock considered to extend to 

 infinity in all directions, thus essentially conforming to Fourier's 

 hypothesis of an infinite solid in the form of a flat plane. 



Reasons have been assigned for doubting the great age sometimes 

 ascribed to the earth, and it might seem like a waste of effort to draw 

 the curves for the Fourier solutions in the case of such immense 

 periods as 1000 million years; but in view of the great uncertainty 

 heretofore prevailing in regard to the age of the consolidation it 

 appeared advisable to conduct the investigation on the broadest lines. 

 The leading characteristics of these Fourier solutions are shown by 

 the rate of increase of temperature downward, and by the depth at 

 which high temperature is attained. The horizontal scale, repre- 

 senting the depth, is the same in all the curves shown in the diagram ; 

 and thus a direct comparison of the effects of the three different 

 periods is possible. In the case of the 1000 million year period the 

 cooling has extended to great depth, more than 445 miles, which 

 exceeds one tenth of the radius ; in fact the increase continues down- 

 ward to about 570 miles, or nearly 0.15 of the radius; but the change 

 of temperature so deep down is excessively slow. 



The curves for icxd million years after the initial epoch are 

 naturally much steeper than those just considered, and accord closely 

 with those drawn by Lord Kelvin, who used the same period. In 

 the solution for the interval of 10 million years after the cooling 

 began the curves are naturally very much steeper yet. In fact for 

 this short period the cooling has not yet extended much lower than 



