REPORT ON CERTAm BRANCHES OF ANALYSIS. 25S 



-re 

 2 



tional. If we successively replace, therefore, a" by -^ + t and 



— — x,yvQ shall get 



•It X 1 . ^ 1 „ 



rir + -J- + "o = cos X + -^ &\a'ax — -^ cos 3 x 

 — :j- sin 4 X + &c. 



/«• + -; 7r- = COS a: yr sin2x ^ cos 3 jr 



4 2 » 3 



H — J- sin 4 d7 + &c. 



Adding these two series together and dividing by 2, we get 



(f J- f^\ •B' 1 1 



^-^ — '- * + -J- = cos ^ IT- cos 3 a: + -^ cos 5 X — &c. {2.) 



It It 

 If X be included between g- and —, then r = and r' = 0, 



and we get 



-r = COS X ^ cos S X + -p- cos 5 j: — &c. (3.) 



4 3 5 ^ •' 



■jt 3 iz* 



If J? be included between -g- and -^, then r = — 1 and r' = 0, 



and we get 



w 1 1 



— -r = cos j: -^ cos 3 ^ + ^r cos 6x — &c. (4.) 



4 3 5 ^ ' 



If the limits of x be -^ and -^, —5- and -^, -^ and 



— -3-, ^ and — '-^, we shall obtain values of the series 



nt Tt 



(2.), which are alternately -7- and — -r-. 



X . 1 . 



Again, if in equation (1.), or rir + -tj- = sin x — -^sinS^r 



+ -q- sin S X -7- sin 4 t + &c., we replace x by tt — x, we 



shall get 



»•' ff -I -^ — = sin or + -^ sin 2 x + -q- sin 3 J? + -j- sin 4 j? + &c. 



Adding these equations together and dividing by 2, we get 



