190 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



Table 4 is a continuation of table 3, giving in terms of the same argu- 

 ment the specific energy of impact for a particle falling from infinity, in 

 billions of ergs per gram, the same in centigrade degrees under specific heat 

 0.2, the surface parabolic velocity in miles per second, and the distortion 

 factor indicating the permanent deformation of elements of the mass at the 

 various points along the radius. 



THE THERMAL PROBLEM. 



It has been supposed in the foregoing that the distribution of heat was 

 not sensibly affected by conduction during the relatively short epoch of 

 accretion. If, now, it be supposed that the subsequent changes in distri- 

 bution are determined by conduction only, in accordance with Fourier's 

 laws, then the form of the temperature curve d = d(r,t) at each instant 

 may be determined by the differential equation 



provided the form of the curve at the initial instant, say t = 0, appropriate 

 conditions relating to the surface, and the values of the conductivity k and 

 specific heat o be assigned. Since the variations of X and o under changes 

 in the physical condition of the substance are almost purely matters of 

 conjecture, the chief value of such an inquiry might well be considered to 

 lie in the determination of features of the thermal phenomenon which seem 

 to persist under varied assumptions on these points. Since, however, the 

 method of superposition of special solutions is practically the only known 

 way of obtaining general solutions of equations like (52), it will be supposed 

 that the latter is linear, A and o being assigned in each special case as func- 

 tions of r but independent of 6, and that the surface equations are linear 

 and homogeneous in 6 and its derivatives. 



According to Fourier's method, of expansion into an infinite series each 

 term of which is a solution of (52) and satisfies the surface condition, the 

 solution may then be sought in the form 



n=l 



where 



and t/„ (x) or y {fi^, x) for n = 1 . . . oo are the appropriate fundamental 

 functions, which are to be determined from 



where 4){x)=XjX^, <p{x) =oplaQPQ, and the successive values of /i employed 

 are those which allow the individual terms of (53) to satisfy the surface 

 condition. The coefficients in (53) will be given by 



£ x^ f {x) F {x) i/„ {x) dx .gg^ 



Cx'(p{x)yr,{xfd^ 



Ar.=d, 



