SECT. IV.] DEVELOPMENT IX SERIES OF FUNCTIONS. 441 



424. We have yet to regard the same propositions under 

 another aspect. If we compare with each other the solutions 

 relative to the varied movement of heat in a ring, a sphere, a 

 rectangular prism, a cylinder, we see that we had to develope 

 an arbitrary function f(x) in a series of terms, such as 



i&amp;lt;/&amp;gt; OvO + &amp;lt;v (/v*0 + 3&amp;lt;!&amp;gt; (/vO + &c - 



The function (, which in the second member of equation 

 (A) is a cosine or a sine, is replaced here by a function which 

 may be very different from a sine. The numbers fi lt //, 2 , //, 3 , &c. 

 instead of being integers, are given by a transcendental equation, 

 all of whose roots infinite in number are real 



The problem consisted in finding the values of the coefficients 

 a \&amp;gt; a v a s - - - a i I they nav e been arrived at by means of definite 

 integrations which make all the unknowns disappear, except one. 

 We proceed to examine specially the nature of this process, and 

 the exact consequences which flow from it. 



In order to give to this examination a more definite object, 

 we will take as example one of the most important problems, 

 namely, that of the varied movement of heat in a solid sphere. 

 \Ve have seen, Art. 290, that, in order to satisfy the initial dis 

 tribution of the heat, we must determine the coefficients a l} a a , 

 r/- s ... a i? in the equation 



ocF(x) = a t sin (^x) + a 2 sin (JLL^X) -4- a 3 sin (p 3 x) + &c ....... (e). 



The function F(x) is entirely arbitrary ; it denotes the value 

 v of the given initial temperature of the spherical shell whose 

 radius is x. The numbers /^, /z- a ... p. are the roots /^, of the 

 transcendental equation 



X is the radius of the whole sphere; h is a known numerical co 

 efficient having any positive value. We have rigorously proved in 

 our earlier researches, that all the values of fju or the roots of the 

 equation (/) are real 1 . This demonstration is derived from the 



1 The Mfrnoircs de V Academic des Sciences, Toine x, Paris 1831, pp. 119 146, 

 contain Rcmarqiifs fjcncralc* sur V application des principes dc Vanalyse algebriquc 



