CRITICAL AND SUPPLEMENTARY. 217 



surface a representation in series ascending powers of s = fj — r, in the form 



p=Pi+aiS + a2S^+ .... y=^i+&iS+M'+ .... 



<T =flri + CjS + CjS^ + .... X=Xi + diS + d2S^+ .... 



such that a few terms suffice to give all the precision which has useful mean- 

 ing under the circumstances. From these come 



P=9iPiS+-2^9iai+Pibi)s^+ .... 

 P _9i^ , pA-^9iai „2 , 



e= f\fds = '^^s^+ .... 

 / p^ as 2pi 



so that the temperature-curve in the neighborhood of the surface has the 

 form 



17=08 "T" . • • • O =x ^ 



2pi OyJ 



and is consequently tangent to the x-axis at the surface-point x = \. The 

 tangency is of ordinary parabolic type, since the vanishing of Oj would mean 

 that the surface material was incompressible. The temperature at first 

 changes at rate 



— .=—._._( Xr^-^r- 1 = + terms with factor {r—r{) 



dt op r dr\ drj a^pi 



Since the first term here is essentially positive, there is necessarily a region 

 just below the surface where the temperature rises. 



On the other hand, at the center there occur maxima of the curves for 

 p, p, and consequently for e. This would most probably happen also for 

 6, so that the appropriate expansion would be of the form 



e=d^-Cr^ .... 



with which the initial rate of change is 



-— - = 5 4- terms with factor r^ 



dt Oopo 



the first term of which is essentially negative, so that the temperature 

 falls in the neighborhood of the center. The only case in which this would 

 not happen would be when o would have a maximum at the center, strong 

 enough to throw the maximum of temperature to a point further out, 

 which is highly improbable, a necessary condition for this being that with 

 expansions of the type 



/)=(Oo(l~6,r2 . . . .) 0=0^(1- CiT^ . . . .) 



C V 



the coefficients satisfy the inequality -A- ^ — ^ 



Ol Po^9 



