. ^ w * ^&amp;lt;e t , 



rv, . 



CHAPTER VI. 



OF THE MOVEMENT OF HEAT IN A SOLID CYLINDER. 



306. THE movement of heat in a solid cylinder of infinite 

 length, is represented by the equations 



dv _ K (d*v ldv\ j A. T/_L ^- n 

 dt ~ CD (dtf + x d~x) l K V h ~dx 



which we have stated in Articles 118, 119, and 120. To inte 

 grate these equations we give to v the simple particular value 

 expressed by the equation v = ue~ mt ; m being any number, and 



jr 



u a function of x. We denote by k the coefficient - which 



enters the first equation, and by h the coefficient -^ which enters 



the second equation. Substituting the value assigned to v, we 

 find the following condition 



m d z u 1 du 



7- -j-j ~ -j- 

 fc axr x ctx 



Next we choose for u a function of x which satisfies this 

 differential equation. It is easy to see that the function may 

 be expressed by the following series ^ 3 



gx* 



./ - 1 _ __ I 



2 



I X-rn 



qy* 



g denoting the constant -r . We shall examine more particularly 



in the sequel the differential equation from which this series 



192 



