Periodicities when the Periods are unknown. 363 



mation in Trigonometry that the snm of n terms of the sines of 

 angles which are in arithmetical progression from the first term of the 



/ t-t , n — \n \ • Ill) 



series (3) written below is -ztsin - , and that begin- 



sin - 

 2 



nb 



sin 



so on. 



f n—1 "1 s ™ ~2~ 

 ning at the second term it is u sin I (U + b) + — — b > and 



sin- 

 2 



The object of the following investigation is to find what relations 

 exist between the series (3) and the series derived from it, in the 

 manner that we are going to describe below. 

 Let the proposed series be 



^{sinU + sin(U + &) + sin(U + 26) + sin(U + 36) . . . &c.} . (3), 

 from which we see that the mean of n terms from the first term of the 



nb 



( tt , n — 1,\ • nl 



series is wsin — , and that the mean of n terms from 



. o 



n sm - 

 2 



the second term of the series is — 



lb 

 2 



{(U + 5) + ^) si n| 



. b 



n sm- 

 2 



or, by reducing, we obtain 



sm 

 2 



. b 



n sin - 

 2 



and similarly the mean of n terms from the third term of the series 

 ib 

 ~V 



n sm 



be— 



is u sin ^Tj+ ^"^^ 6^- ^ n n \ a .. ._ 



n sm— 

 2 



u\ sm 



. nb nb 

 / n-1 \ sm "2" / n+l \ sin Y 



n sin- n sin- 



2 2 



. nb 



sin-^r 



+ sin(uV-^) 2 r + ^ &c -> ■ 



