﻿134 Rev. 0. Fisher on the Level of 



This differs from the expression for the corresponding time 

 rate (dv/dt) in the case when sphericity is neglected by being 

 multiplied by the factor r/(r—x). Hence the fall of tem- 

 perature at a given depth goes on slightly more rapidly in 

 the case of the sphere, as might be expected. 



Again, differentiating with respect to x, we have 



du _ r y JJ_ 1 -^i 

 dx t—x *fijr x/4jci 



{r—xf \ >Jtt V4«U« ' / 



Putting x = 0, we get for the temperature gradient at the 

 surface 



\ Vtt \Z±Kt rf 

 whence 



. 2 1 



^ = ^r III ' 



When sphericity is neglected, or r infinite, we have 



2 Y 

 J± K l — — -= — , = a of Lord Kelvin's problem of secular 



cooling, and as in my ' Physics &c.' Hence the time which 

 elapses before a given surface-temperature gradient is ac- 

 quired is somewhat shorter when sphericity is taken account 

 of, as might also have been expected. 



Lord Kelvin assumed the high value of 7000° F. for V, 

 the temperature of solidification, probably to allow for its 

 possible increment in the lower shells owing to the pressure. 

 With this value a = 402,832 feet. But, when sphericity is 

 taken account of, sjkict becomes 396,073 feet, which makes 

 it 1 mile and 1497 feet less. 



The equation which gives x, the depth of the level of no 

 strain, is * — 



Se C r , . 2 du , , . du 



If we substitute the value just found for du/dt, and take 

 */4/ct for the unit of length, this may be reduced to the 



* ' Physics, &c.,' p. 95. 



