230 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



SO that as v decreases from the value v^, the sign of (p' is ultimately that of 

 oA - — ^, unless this be zero. If then ^ is to be an increasing function of 

 V, the most moderate value of A which can be assumed is 



-H^V^A (167) 



which as appears from (22) and (58) would give from the first term of (163) 

 a temperature distribution precisely the same also in absolute values as 

 that deduced under the first hypothesis in Part I. 



This result shows that there is not necessarily any radical difference 

 between the temperature-distributions deducible from the two alternative 

 extreme suppositions described, though variations in the special hypotheses 

 may be expected to produce here, as under the opposite view, considerable 

 differences in the quantitative conclusions. 



It has been pointed out that the surface-gradient of temperature as 

 deduced from the theory would probably never reach the value at present 

 found by observation, at least unless the conductivity increase consider- 

 ably from the surface downward. But the present theory has taken ac- 

 count only of the heat obtained from gravitational energy, to which must 

 be added that obtained from several other assignable sources. These might 

 be relatively unimportant, so that their effect could be included in the 

 sense of corrections to the above results; but if of sufficient magnitude they 

 might alter the features of the thermal process completely. 



In particular, if heat were steadily generated in each unit of volume at 

 rate (fir), the equation of conduction would be 



,p^^\l-Ur^^)+^(r) (168) 



^ dt r^ dr\ drj 



the solution of which may be written 



» = „+ /''* /'.VW* (169) 



'-'*£H 



where u satisfies the equation 



and is to be determined so that 6 satisfies the given initial and boundary 

 conditions. For example, if (p is constant or the rate of generation per 

 unit volume the same throughout the mass, the steady component, defined 

 by the integral in (169), is, with X constant: 



whence ^^ /? _^ k=^ 



so that with the flux corresponding to thermal equilibrium a surface gra- 

 dient of 1° in 30 meters v/ould mean a central temperature of 100,000°, 



