SECT. I.] ULTIMATE LAW OF TEMPERATURE. 277 



1st, The partial differential equation is satisfied ; 2nd, the quantity 

 of heat which escapes at the surface accords at the same time with 

 the mutual action of the last layers and with the action of the air 



on the surface ; that is to say, the equation -?- + hx = 0, which 



each part of the value of v satisfies when x X, holds also when 

 we take for v the sum of all these parts ; 3rd, the given solution 

 agrees with the initial state when we suppose the time nothing. 



292. The roots n lt n 2 , 7? 3 , &c. of the equation 



nX _, ,_ 



7 V&quot; 1 /&-A. 



tan n X. 



are very unequal; whence we conclude that if the value of the 

 time is considerable, each term of the value of v is very small, 

 relatively to that which precedes it. As the time of cooling 

 increases, the latter parts of the value of v cease to have any 

 sensible influence ; and those partial and elementary states, which 

 at first compose the general movement, in order that the initial 

 state may be represented by them, disappear almost entirely, one 

 only excepted. In the ultimate state the temperatures of the 

 different layers decrease from the centre to the surface in the 

 same manner as in a circle the ratios of the sine to the arc 

 decrease as the arc increases. This law governs naturally the 

 distribution of heat in a solid sphere. When it begins to exist, 

 it exists through the whole duration of the cooling. Whatever 

 the function F (x) may be which represents the initial state, the 

 law in question tends continually to be established ; and when the 

 cooling has lasted some time, we may without sensible error 

 suppose it to exist. 



293. We shall apply the general solution to the case in 

 which the sphere^ having been for a long time immersed in a 

 fluid, has acquired at all its points the same temperature. In 

 this case the function F(x) is 1, and the determination of the 

 coefficients is reduced to integrating x sin nx dx, from x = to 

 x = X : the integral is 



sin nX nX cos n X 



