SECT. II.] DIFFERENT REMARKS OX THIS SOLUTION. 279 



SECTION II. 

 Different remarks on this solution. 



294&amp;lt;. We will now explain some of the results which may be 

 derived from the foregoing solution. If we suppose the coefficient 

 h, which measures the facility with which heat passes into the air, 

 to have a very small value, or that the radius X of the sphere is 

 very small, the least value of e becomes very small ; so that the 



, 



- h v . , , , 

 equation - = 1 -^ X is reduced to - = - = 1 



e -273 63 



ohX 

 or, omitting the higher powers of e, e 2 = ^- . On the other 



hand, the quantity - -- cos e becomes, on the same hypothesis, 



. ex 



27 Y Sm X 



^ And the term is reduced to 1. On making these 

 K ex 



X 



_ 8fr t 



substitutions in the general equation we have v = e Ci)X -f &c. 

 We may remark that the succeeding terms decrease very rapidly 

 in comparison with the first, since the second root n 9 is very much 

 greater than ; so that if either of the quantities h or X has 

 a small value, we may take, as the expression of the variations 



Sht 



of temperature, the equation v = e 67&amp;gt;j: . Thus the different 

 spherical envelopes of which the solid is composed retain a 

 common temperature during the whole of the cooling. The 

 temperature diminishes as the ordinate of a logarithmic curve, the 

 time being taken for abscissa ; the initial temperature 1 is re- 



_ * h A. 

 duced after the time t to e C DX . In order that the initial 



temperature may be reduced to the fraction , the value of t 



Y 



must be ^y CD log m. Thus in spheres of the same material but 



