CRITICAL AND SUPPLEMENTARY. 205 



Table 9, based on the formula p=Po(l-p^), gives the values at va- 

 rious depths of the density, of the pressure and bulk-modulus in millions of 

 atmospheres, and the thermal measure for = 0.2 of the specific compres- 

 sional energy. 



To estimate the modifications in the thermal process due simply to the 

 change in the formula for density, it will be well to repeat the former alter- 

 native hypotheses regarding the specific heat. 



If a, k are constant, the primitive temperature-curve by (78) is 



e=d,(i-xr{i-jx^)^(i-^x^) e,= '^j (82) 



which is tabulated in column 5 of table 9, while equation (55) reduces to 



giving for the fundamental functions 



y=^{-iya,x'' (84) 



where the coefficients a^ are determined successively from 



«o = l «i=f ''^ = 2i{2l+r)^'''-''^'''''-'^ t = 2 . . . .00 (85) 



with the values of // which make y vanish at re = 1. The complete computa- 

 tion to sufl&cient accuracy to determine the details of the conduction is 

 again hardly practicable, but the first two fundamental functions with their 

 parameters fx, r, and coefficients A determined from (56) are: 



yi = l- 2.04360 x^ + 1 .69080 x*- .91977 rc« + .36231 a;^- .1 1362 x^»^ 

 + .02927 x'' - .00645 x'' + .00123 x'^ - .00021 x^* 

 + .00003x2"- _ 



/ii = 12.2616 ^1 = 1.089^0 Ti =400 billion years 



^2 = 1-8.59393x2 + 23.99826.1;''- 36.99908 x« + 38.77346 x« i /gg) 



-30.56380 x»'' + 19.25673 x»2- 10.08879 xi^ + 4.52008 x^« 

 - 1.76799 xi8 + .61344 x-"- .19120 x" + .05409 x^" 

 -.01400 .x''' + . 00334x28 -.00074x3" + .00015x^2 

 -.00003 .x3'*+ .... 



//j = 51.5636 A2 = -.0705 (?„ T2=96 billion years 



These are given in table 10, columns 2 and 3, together with the residuals 

 from the primitive curve due to components higher than the first and second 

 respectively in columns 4 and 5, the tabulation in all cases using as unit the 

 primitive central temperature d^, which for (7 = 0.2 is about 16,600°. The 

 last column of that table gives the initial rate of change of temperature, 

 computed from 



dd 

 dt 



dA ( 312 , 1268 , 132 , , ,_ , 100 ^l . L 5 )' .o^x 



=^l-^+-49-^-^^+^'^-l9-^7"i'"7-^l ^''^ 



