SECT. II.] RADIUS OF SPHERE VERY GREAT. 287 



We have, therefore, if we denote the mean temperature by z, 



f - \o *K&amp;lt;iH , . N2 Kcft 



= (sm 6, - ^ cos ej 2 -fitx* , (sm e, - 6 2 cos e g ) -^P 

 .4 e 3 26 - sin 2e * 6 3 2e - sin 2e 



an equation in which the coefficients of the exponentials are all 

 positive. 



302. Let us consider the case in which, all other conditions 

 remaining the same, the value X of the radius of the sphere 

 becomes infinitely great 1 . Taking up the construction described 



r &quot;F&quot; 



in Art. 285, we see that since the quantity ^- becomes infinite, 



the straight line drawn through the origin cutting the different 

 branches of the curve coincides with the axis of x. We find then 

 for the different values of e the quantities TT, 2?r, Sir, etc. 



_A !i!&amp;lt; 

 Since the term in the value of z which contains e CD x * 



becomes, as the time increases, very much greater than the 

 following terms, the value of z after a certain time is expressed 



J--T o 



without sensible error by the first term only. The index -^= 



CD 



KTT Z 

 being equal to 7 ^y a , we see that the final cooling is very slow 



in spheres of great diameter, and that the index of e which 

 measures the velocity of cooling is inversely as the square of the 

 diameter. 



303. From the foregoing remarks we can form an exact idea 

 of the variations to which the temperatures are subject during the 

 cooling of a solid sphere. The initial values of the temperatures 

 change successively as the heat is dissipated through the surface. 

 If the temperatures of the different layers are at first equal, or 

 if they diminish from the surface to the centre, they do not 

 maintain their first ratios, and in all cases the system tends more 

 and more towards a lasting state, which after no long delay is 

 sensibly attained. In this final state the temperatures decrease 



1 Biemann has shewn, Part. Diff. gleich. 69, that in the case of a very large 

 sphere, uniformly heated initially, the surface temperature varies ultimately as the 

 square root of the time inversely. [A. F.] 



