SECT. VI.] DEVELOPMENT IN SERIES OF COSINES. 191 



taken from x = to x IT, supposing j and i to be integers. We 

 have 



This integral, taken from x = to x TT, evidently vanishes 

 whenever j and i are two different numbers. The same is not 

 the case when the two numbers are equal. The last term 



sn - 



becomes ~ ,| and its value is \TT, when the arc x is equal to 77% 



If then we multiply the two terms of the preceding equation (m) 

 by cos ix, and integrate it from to TT, we have 



&amp;lt;/&amp;gt; (X) COS IX dx = ^TTdi, 



an equation which exhibits the value of the coefficient c^. 



To find the first coefficient , it may be remarked that in 

 the integral 



i t 



dn (ji) x, 



if j = and i = each of the terms becomes ^ , and the value 



of each term is JTT ; thus the integral I cos jx cos ix dx taken 



from x = to x = TT is nothing when the two integers j and i 

 are different : it is \tr when the two numbers j and i are equal 

 but different from zero ; it is equal to TT when j and i are each 

 equal to zero ; thus we obtain the following equation, 



1 f v [&quot; fir 



2 Jo Jo Jo 



+ cos 3# I &amp;lt;/&amp;gt; (a?) cos 3# d# + &c. (n)\ 



J o 



1 The process analogous to (A) in Art. 222 fails here ; yet we see, Art. 177, that 

 an analogous result exists. [B. L. E.] 



