r 

 I 



448 THEORY OF HEAT. [CHAP. IX. 



which only odd powers enter, is resolved, Art. 225, into a series 

 of cosines which contain only even powers. 



In the actual problem relative to the sphere, the value of 

 xF(x) is developed by means of equation (e). We must then, 

 as we see in Art. 290, write in each term the exponential factor, 

 which contains t, and we have to express the temperature v, 

 which is a function of x and t, the equation 



x 

 dxsin (fai) aF(ca) 



.. ...... (E). 



sin (/i 4 0) sin fo/3) 



The general solution which gives this equation (E} is wholly 

 independent of the nature of the function F(x) since this function 

 represents here only an infinite multitude of arbitrary constants, 

 which correspond to as many values of x included between 

 and X. 



If we supposed the primitive heat to be contained in a part 

 only of the solid sphere, for example, from x = to x = $X, 

 and that the initial temperatures of the upper layers were nothing, 

 it would be sufficient to take the integral 



sin (^a )/(), 



between the limits x = and x = ^X. 



In general, the solution expressed by equation (E) suits all 

 cases, and the form of the development does not vary according to 

 the nature of the function. 



Suppose now that having written sin x instead of F(x) we have 

 determined by integration the coefficients a t) and that we have 

 formed the equation 



x sin x = a t sin JJL^X + 2 sin JJL Z % + a 3 sin JJL^X -f &c. 



It is certain that on giving to x any value whatever included 

 between and X, the second member of this equation becomes 

 equal to a; since; this is a necessary consequence of our process. 

 But it nowise follows that on giving to a; a value not included 

 between and X, the same equality would exist. We see the 

 contrary very distinctly in the examples which we have cited, and, 



