310 THEORY OF HEAT. [CHAP. VI. 



From the general solution we might deduce consequences 

 similar to those offered by the movement of heat in a spherical 

 mass. We notice first that there are an infinite number of 

 particular states, in each of which the ratios established between 

 the initial temperatures are preserved up to the end of the cooling. 

 I When the initial state does not coincide with one of these simple 

 I states, it is always composed of several of them, and the ratios of 

 the temperatures change continually, according as the time increases. 

 In general the solid arrives very soon at the state in which the 

 temperatures of the different layers decrease continually preserving 

 the same ratios. When the radius X is very small 1 , we find that 



2ft 



the temperatures decrease in proportion to the fraction e&quot; CDX. 



If on the contrary the radius X is very large 2 , the exponent of 

 e in the term which represents the final system of temperatures 

 contains the square of the whole radius. We see by this what 

 influence the dimension of the solid has upon the final velocity of 

 cooling. If the temperature 3 of the cylinder whose radius is X y 

 passes from the value A to the lesser value B, in the time T, the 

 temperature of a second cylinder of radius equal to X will pass 

 from A to B in a different time T . If the two sides are thin, the 

 ratio of the times T and T f will be that of the diameters. If, on 

 I the contrary, the diameters of the cylinders are very great, the 

 1 ratio of the times T and T will be that of the squares of the 

 diameters. 



1 When X is very small, Q = -% &amp;gt; from tlie equation in Art. 314.^ Hence 



_ &kt e 2hM 



e ^ becomes e, ^ . 

 In the text, h is the surface conducibility. 



2 &quot;When X is very large, a value of B nearly equal to one of the roots of the 



B B B fi 

 quadratic equation 1= _ w ill make the continued fraction in Art. 314 



i O 4 O 



assume its proper magnitude. Hence 0=1-446 nearly, and 



_?2to0 5 78 ft* 



e, x * becomes e x * . 

 The least root of /(0) = is 1-4467, neglecting terms after 4 . 



3 The temperature intended is the mean temperature, which is equal to 



[A. P.] 



