184 THEORY OF HEAT. [CHAP. III. 



Collecting the coefficients of sin x, sin 2x, sin 3#, sin 4*x, &c., 

 I.i 

 have 



and writing, instead of * + -* 7+ etc.. its value -, - , we 



n n* n 5 tf ri* + 1 



1 (e* e x ) _ sin x sin 2x sin 3# 



2 71 &quot; e^-e^ ~1~11~ 



We might multiply these applications and derive from them 

 several remarkable series. We have chosen the preceding example 

 because it appears in several problems relative to the propagation 

 of heat. 



219. Up to this point we have supposed that the function 

 whose development is required in a series of sines of multiple 

 arcs can be developed in a series arranged according to powers 

 of the variable x t and that only odd powers enter into that 

 series. We can extend the same results to any functions, even 

 to those which are discontinuous and entirely arbitrary. To esta 

 blish clearly the truth of this proposition, we must follow the 

 analysis which furnishes the foregoing equation (B), and examine 

 what is the nature of the coefficients which multiply sin a?, 



sin 2x, sin 3#, &c. Denoting by - the quantity which multiplies 



ftr 



-sin nx in this equation when n is odd, and s mnx when n is 

 n n 



even, we have 



a = &amp;lt;KT) - J *&quot; + J &amp;lt;f W - i * + &C. 



Hi Hi It/ 



Considering s as a function of TT, differentiating twice, and 



1 d?s 

 comparing the results, we find s + -$ ~r- 2 = &amp;lt;/&amp;gt; (TT) ; an equation 



ft Cv r /r 



which the foregoing value of 5 must satisfy. 



1 d z s 



Now the integral of the equation s +-5 T~I = &amp;lt;/&amp;gt; (#)&amp;gt; m which s 



f ft CLtjG 



is considered to be a function of a?, is 

 s a cos nx + b sin nx 



4- n sin nx \ cos nx $ (x) dx n cos nx I sin nx (x) dx. 



