SECT. II.] VARIED MOVEMENT IN A SPHERE. 93 



must be made to become nul, when we give to x its complete 

 value X equal to the radius of the sphere, whatever else the value 

 of t may be. We should then have, on this hypothesis, if we 

 denote by &amp;lt;f&amp;gt; (x, t) the function of x and t, which expresses the 

 value of v, the two equations 



jr = -F^ ( -T- 2 + - 3- ) , and 6 (X, t) = 0. 

 dt \jj-J \(zx x cl/jcj 



Further, it is necessary that the initial state should be repre 

 sented by the same function &amp;lt; (x, t) : we shall therefore have as a 

 second condition (/&amp;gt; (x, 0) = 1. Thus the variable state of a solid 

 sphere on the hypothesis which we have first described will be 

 represented by a function v, which must satisfy the three preceding 

 equations. The first is general, and belongs at every instant to 

 all points of the mass ; the second affects only the molecules at 

 the surface, and the third belongs only to the initial state. 



115. If the solid is being cooled in air, the second equation is 

 different ; it must then be imagined that the very thin envelope 

 is maintained by some external cause, in a state such as to pro 

 duce the escape from the sphere, at every instant, of a quantity of 

 heat equal to that which the presence of the medium can carry 

 away from it. 



Now the quantity of heat which, during an infinitely small 

 instant dt, flows within the interior of the solid across the spheri 

 cal surface situate at distance x, is equal to 4&amp;gt;K7rx z -^- dt ; and 



this general expression is applicable to all values of x. Thus, by 

 supposing x = X we shall ascertain the quantity of heat which in 

 the variable state of the sphere would pass across the very thin 

 envelope which bounds it ; on the other hand, the external surface 

 of the solid having a variable temperature, which we shall denote 

 by F, would permit the escape into the air of a quantity of heat 

 proportional to that temperature, and to the extent of the surface, 

 which is 4&amp;lt;7rX 2 . The value of this quantity is 4&amp;lt;h7rX 2 Vdt. 



To express, as is supposed, that the action of the envelope 

 supplies the place, at every instant, of that which would result from 

 the presence of the medium, it is sufficient to equate the quantity 



4&amp;gt;JnrX*Vdt to the value which the expression 4iK TrX* -_,- dt 



