192 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



that not all of the energy can be considered as manifested in the raising 

 of the temperature, this number is probably somewhat too high. But in 

 any case it appears that the planetesimal hypothesis could thus plausibly 

 assign an origin for precisely the kind of initial thermal condition of the 

 mass postulated in Kelvin's famous theory, which would then serve to 

 indicate the subsequent thermal process, and in particular the probable 

 age of the earth reckoned from the close of the epoch of formation. 



It seems, however, more in accord with the spirit of the general hypoth- 

 esis to assign no such relative importance to the energy of impact. For 

 the value of e^ should probably be assumed much less than its parabolic 

 value, being due to the relative velocities simply of bodies which might 

 partake more or less of a common motion; for instance of bodies moving 

 in similar directions around the sun in intersecting orbits of moderate eccen- 

 tricity. Moreover, even with what might be considered a rapid rate of 

 accretion, when compared with astronomical processes in general, by far the 

 greater part of the heat so generated must be expected to escape quickly 

 by radiation, thus making v a small fraction. The computations following 

 refer to the case when the value of ve^ is insignificant, so that the compres- 

 sional energy alone is effective in producing the primitive temperatures, 

 which are then to be computed from column 6 of table 2 and the assumed 

 values of o. 



The same relative freedom of dissipation into space would tend to keep 

 the surface at a low and equable temperature after the accretion had 

 ceased. It will, therefore, be assumed that the surface temperature keeps 

 a fixed value, which may be taken as the zero of reckoning, and whose 

 place in an absolute scale would depend largely on the thermal influence 

 of the atmosphere. It is further supposed that the rise of temperature 

 produced by compression can be reckoned from this point. With these 

 stipulations the solution will be determined if definite hypothesis is made 

 regarding the conductivity and specific heat. 



If >l and o are uniform throughout the mass, then <p{x)=l, and <p{x) 

 = plPo' Since great precision is needless in these computations, the ratio 

 ^/iOo will be replaced by the convenient expression (1-^x^y, whose close 

 accordance with it appears from columns 4 and 5 of table 1, and which 

 simplifies the determination of the solutions of equation (55), which then 

 reduces to 



the required solution of which may be written: 



2/(/i,a:)=T(-ir«i^" (60) 



1=0 



where the a's, which are functions of /x, are all positive and determined by 

 the simple recursion-formula 



^i = 2i(2i + i) (^^-i + ^^-2 + i«»-3) «o = l «.=! (^^^ 



