198 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



and for the expansion (53) the coefficients: 





A„ = 2n;r / xF (x) sin mix dx (68) 



if the initial temperature-curve is y = d^F{x) and the coefficients are ex- 

 pressed in terms of d^ as unit. The intervals z are determined by 



and are thus in this case strictly proportional to the squared reciprocals of 

 the natural numbers. 



To determine F{x), it may be noted that the initial temperatures can be 

 determined from those of the previous hypothesis by multiplying by pjpi, 

 giving for the central temperature 



which for <t = 0.2 is about 81,000°; and for the curve in terms of the central 

 ordinate 



^ F^^)JPlZPl) (71) 



For convenience in determining the coefficients A^ the last equation will 

 be replaced by the formula 



F{x)={\-x''y{l-jX-'-Vjx') 



or 



F{x) =l_^x^ + ||a;^-|a;« + |:c'' (72) 



which, in comparison with (71), leaves residuals at most 3 units in the 

 fourth decimal place. By (68) the expansion has then the coefficients 



where 



2n\—S W-— ,<? (2)_|__^,Q (3)__^^ w_L_L^ (5)1 



J ' 



which can be reduced through integration by parts, giving finally 



4.=2(-l)-f-=l?^,-#?-.-?f-? + ^| (73) 



The first few of these coefficients are : 

 A,= +0.70590 ^2- +0.36797 ^4= +0.05354 ile= +0.02240 

 vlg^ -0.10765 Ag= -0.03284 ^7= -0.01632 



