280 THEORY OF HEAT. [CHAP. V. 



of different diameters, the times occupied in losing half or the 

 same defined part of their actual heat, when the exterior con- 

 ducibility is very small, are proportional to their diameters. The 

 same is the case with solid spheres whose radius is very small ; 

 and we should also &quot;find the same result OB attributing to the 



interior conducibility K a very great value. The statement holds 



7 ~y 

 generally when the quantity -^ is vejy small. , We may regard 



the quantity ^ as very small when the body which is being 



cooled is formed of a liquid continually agitated, and enclosed in 

 a spherical vessel of small thickness. The hypothesis is in some 

 measure the same as that of perfect conducibility; the tem 

 perature decreases then according to the law expressed by the 



Sht 



equation v = e C1JX . 



295. By the preceding remarks we see that in a solid sphere 

 which has been cooling for a long time, the temperature de 

 creases from the centre to the surface as the quotient of the sine 

 by the arc decreases from the origin where it is 1 to the end 

 of a given arc e, the radius of each layer being represented 

 by the variable length of that arc. If the sphere has a small 

 diameter, or if its interior conducibility is very much greater 

 than the exterior conducibility, the temperatures of the successive 

 layers differ very little from each other, since the whole arc e 

 which represents the radius X of the sphere is of small length. 

 The variation of the temperature v common to all its points 



Sht 



is then given by the equation v e cux . Thus, on comparing the 

 respective times which two small spheres occupy in losing half 

 or any aliquot part of their actual heat, we find those times 

 to be proportional to the diameters. 



_ 3M 



296. The result expressed by the equation v = e CDX belongs 

 only to masses of similar form and small dimension. It has been 

 known for a long time by physicists, and it offers itself as it were 

 spontaneously. In fact, if any body is sufficiently small for the 

 temperatures at its different points to be regarded as equal, it 

 is easy to ascertain the law of cooling. Let 1 be the initial 



