342 THEORY OF HEAT. [CHAP. IX. 



Let ~Y r, and denote F (x) or F(- j by f(r) ; we have 

 f(r) = j sin r + 2 sin 2r + a a sin 3r -f &c. 



2 r 



Now, we have previously found a = - \drf(r) sinir, the inte 

 gral being taken from r = to r = TT. Hence 



The integral with respect to x must be taken from x = to 

 x = X Making these substitutions, we form the equation 



sin 



353. Such would be the solution if the prism had a finite 

 length represented by 2X. It is an evident consequence of the 

 principles which we have laid down up to this point; it remains 

 only to suppose the dimension X infinite. Let X= UTT, n being 

 an infinite number; also let q be a variable whose infinitely small 



increments dgr are all equal ; we write -7- instead of n. The general 



term of the series which enters into equation (a) being 



. ITTX , .. 



sin -- , 



jpi 2 * . ITTX ( , .. 

 sm^jdxF (x) 



we represent by 3- the number i, which is variable and becomes 

 infinite. Thus we have 



-v IT 1 . q 



JL = -T-, n = -7- , fc=-j-. 

 dy dq dqr 



Making these substitutions in the term in question we find 

 e~ kqH sin gx\dxF (x) sin qx. Each of these terms must be divided 



*7T 



by X or v-, becoming thereby an infinitely small quantity, and 



