292 THEORY OF HEAT. [CHAP. VI. 



is derived; here we consider the function u to be known, and 

 we have ue~ 01ct as the particular value of v. 



The state of the convex surface of the cylinder is subject 

 to a condition expressed by the definite equation 



which must be satisfied when the radius x has its total value X\ 

 whence we obtain the definite equation 



oa 9 a 4 2 9 2 4, 2 fi 2 



2 V * Tl U 



thus the number $r which enters into the particular value ue~ u 

 is not arbitrary. The number must necessarily satisfy the 

 preceding equation, which contains g and X. 



We shall prove that this equation in g in which h and X 

 are given quantities has an infinite number of roots, and that 

 all these roots are real. It follows that we can give to the 

 variable v an infinity of particular values of the form ue~ aM , 

 which differ only by the exponent g. We can then compose 

 a more general value, by adding all these particular values 

 multiplied by arbitrary coefficients. This integral which serves 

 to resolve the proposed equation in all its extent is given by 

 the following equation 



v = a^e ^ 4- a 2 w 2 e~^ w 4- 3 w 3 e~^ 3&amp;lt; + &c., 



ffi&amp;gt; 9v 9a&amp;gt; & Ct denote all the values of g which satisfy the definite 

 equation ; u v u z , u s , &c. denote the values of u which correspond 

 to these different roots; a l9 a z , a a , &c. are arbitrary coeffi 

 cients which can only be determined by the initial state of the 

 solid, 



307. We must now examine the nature of the definite 

 equation which gives the values of g, and prove that all the roots 

 of this equation are real, an investigation which requires attentive 

 examination. 



