226 SEE— TEMPERATURE, SECULAR COOLING [April 20, 



And 

 '^ aX dx ^ a\X dx ^ ~' X dx ^ \ a\ X dx ^ 



- r^^^-d \ -I r^^d - r^^^d'i 



Jo dx J ' ' a\X dx ~Jo dx ^ \' 

 Finally 



e.=e+(@,-0;)+(0;-0;')+(®;'-®;")+ • • • + (0;-'-<^;-)- (32) 



the subscripts on the right corresponding to the i arbitrary epochs. 

 In the application of these equations we must remember that this 

 multiple process is valid only for values of x smaller than that corre- 

 sponding to the maximum of @i. For in Fig. D, we see that there is 

 a maximum to the multiple temperature curve, and the true course 

 of it can not be determined beyond this point. The apparent fall of 

 temperature at greater depths indicates the failure of the process. 

 But by taking suitable periods of time, and appropriate values for 

 V, V, V" y . . . F% one may approximate the true curve asymptotically 

 near the surface, and carry the determination to any desired depth. 



In our present numerical work we have thought it sufficient to 

 take Ti = 10 million years, and F'= V = 2000° . By carrying this 

 process far enough we may obtain a multiple solution which is almost 

 absolutely rigorous, and it will include all the effects of rising tem- 

 perature beneath the surface as well as the uniform temperature 

 embraced in the original solution of Fourier. The principal difficulty 

 in extending the method to the propagation of heat at great depths 

 is the uncertainty respecting the original law of temperature within 

 the earth, before encrustation began. But it appears clearly and un- 

 mistakably that the true curve of temperature is much steeper than 

 that resulting from the simple Fourier solution. At greater depths 

 the steepness increases more and more, till it finally takes the form of 

 the arc of an ellipse, as already pointed out. The curve is thus con- 

 cave near the surface, and convex at great depths, so that it has a 

 point of flexure, probably at no very great depth, but the exact loca- 

 tion cannot be determined. The simple Fourier solutions always 

 make the curves of temperature convex near the surface, which is 

 a serious defect and introduces discontinuity at greater depths. 



It may be noticed that when all the arguments of the times 



