CHAPTER V. 



OF THE PROPAGATION OF HEAT IN A SOLID SPHERE. 

 SECTION I. 



General solution. 



283. THE problem of the propagation of heat in a sphere has 

 been explained in Chapter II., Section 2, Article 117; it consists 

 in integrating the equation 



dv , fd*v 2 dv\ 

 so that when x X the integral may satisfy the condition 



, 

 ax 



k denoting the ratio , and h the ratio -^ of the two con- 



ducibilities ; v is the temperature which is observed after 

 the time t has elapsed in a spherical layer whose radius is a?; 

 X is the radius of the sphere ; v is a function of x and t, which is 

 equal to F (x) when we suppose * = 0. The function F(x) is 

 given, and represents the initial and arbitrary state of the solid. 

 If we make y = vx, y being a new unknown, we have, 



after the substitutions, ^f = ^T^ : tnus we must in t e g r ate the 



last equation, and then take , We shall examine, in the 



sc 



first place, what are the simplest values which can be attributed 

 to if, and then form a general value which will satisfy at the same 



