300 Level of No Strain in a Cooling Homogeneous Sphere. 



appears justifiable, when we reflect that previous to consolidation 

 some cooling of the molten material must have taken place by con- 

 vective currents. I have ventured to slightly expand the beginning 

 of M. Rudski's note, since it seemed to me somewhat too brief. — 



a. h. d.] 



IF v denotes the temperature in the cooling sphere, the 

 equation of cooling is, in Thomson's notation, 



dt 



\dr* r dr/ ^ ' 



Now it may be shown that if K be the modulus of stretching 

 in any stratum, then 



dt r s J ' 



*£h d *> • • • • (2) 



where e is a certain constant, and where the integral is taken 

 from the stratum r down to such a depth that there is no 

 change of temperature *. 



If (2) be integrated by parts, then, on the assumption that 

 at the centre of the sphere there is no change of temperature, 

 we have from (1) 



dK. _ dv 3efc dv 

 dt dt r dr 



ndv_dH\ 



\r dr drV W 



\r dr 



Then the strata are compressed, unstrained, or stretched 

 according as 



dv :> 3k dv 



dt < r dr' 



To show the influence of the initial distribution of tempe- 

 rature, take the following example: 



Let, for simplicity, the unit of length be so chosen that the 

 radius of the sphere is equal to ir. Let the temperature of 

 the external medium be equal to zero. 



(Suppose that the initial temperature decreased from the 

 centre to the surface according to the simple law 



sin r m 



r ~ 9 



r 



then, at any time afterwards, we have for the temperature of 



* Phil. Trans. Roy. Soc. vol. clxxviii. (1887) A. p. 244. 



