1907] 



AND CONTRACTION OF THE EARTH. 



225 



tabulated, and added algebraically to the ordinates of the original 

 Fourier solution, in which the uniform temperature was 2000°, and 

 ^=10 million years, gives very nearly the true curve for the tem- 

 perature near the surface. This is illustrated by Fig. D, in the 

 diagram. The formulae used are : 

 For the rate of augmentation of temperature 



d® 



V - 



dx ZSAVt 



(27) 



where both t and r are taken to be each 10 million years, and V = V . 

 For the curve of temperature we have 







d(d , 



—.- dx 

 dx 



d%' , 



—r dx 

 dx 



(28) 



^ I rd% , I f rd% , rd%' , 1 ^ ,^ ^, , , 



in which a=2i/^ (cf. Appendix D, Thomson & Tait's " Nat Phil.," 



Vol. -L, part II., p. 477). 



More generally the formulae become 



d% I V __^i 

 — = e i6oo< 



d'"^ 35-4 1// 



d%' _ I V 

 dx 



d%" 

 'di 



d^ 

 ~dx 



35'4i/V+Tj 

 I V" 



e i6oo(^+Ti) , 



35.4t//+t^4-t2 

 I V 



e i6oo(<+Ti+To') 



x« 



C 1600(<+T1+T2+ ...+Ti). 



35-4i//+T^ + T2+..- + T^ 

 Or when the times are reckoned from arbitrary epochs 



d^ 



dx 



I V^ • e 1600{<+(Tl-<l) + (T2-^2)+..-+(Ti-<i)] 



35-4i/;+(t,-/,) + (t,-/^+...+ (t.-/.) 



(30) 



