288 THEORY OF HEAT. [CHAP. V. 



from the centre to the surface. If we represent the whole radius 

 of the sphere by a certain arc e less than a quarter of the 

 circumference, and, after dividing this arc into equal parts, take 

 for each point the quotient of the sine by the arc, this system of 

 ratios will represent that which is of itself set up among the 

 temperatures of layers of equal thickness. From the time when 

 these ultimate ratios occur they continue to exist throughout the 

 whole of the cooling. Each of the temperatures then diminishes 

 as the ordinate of a logarithmic curve, the time being taken for 

 abscissa. We can ascertain that this law is established by ob 

 serving several successive values z, z , z&quot;, z &quot; y etc., which denote 

 the mean temperature for the times t, t + , t + 2, t + 3, etc. ; 



the series of these values converges always towards a geometrical 



/ n 



progression, and when the successive quotients -, , , -77-, , etc. 



z z z 



no longer change, we conclude that the relations in question are 

 established between the temperatures. When the diameter of the 

 sphere is small, these quotients become sensibly equal as soon as 

 the body begins to cool. The duration of the cooling for a given 

 interval, that is to say the time required for the mean tem 

 perature z to be reduced to a definite part of itself , increases 

 as the diameter of the sphere is enlarged. 



304. If two spheres of the same material and different 

 dimensions have arrived at the final state in which whilst the 

 temperatures are lowered their ratios are preserved, and if we 

 wish to compare the durations of the same degree of cooling in 

 both, that is to say, the time which the mean temperature 



of the first occupies in being reduced to , and the time in 



m 



which the temperature z of the second becomes , we must 



m 



consider three different cases. If the diameter of each sphere is 

 small, the durations and are in the same ratio as the 

 diameters. If the diameter of each sphere is very great, the 

 durations and are in the ratio of the squares of the 

 diameters; and if the diameters of the spheres are included 

 between these two limits, the ratios of the times will be greater 

 than that of the diameters, and less than that of their squares. 



