Periodicities when the Periods are unknown. 365 



proportional with the proposed series from ^"^"^ th term when n is 

 an odd number, and that the second mean series is proportional with 

 the proposed series from ^- -^ th term when n is an even number. 



8. Suppose in the proposed series that b = — ; then since 



n 



nb . A 

 sm ■ — =sin 7r=0, 

 2 



the mean of the sines of the proposed series will be zero, and conse- 

 quently the first mean series or the second mean series will also be 

 zero, 



. nb . nb b 

 sin sin -g- cos g 



9. The factor or is a proper fraction. 



. b • b 



n sm — n sin — 



2 2 



Put -=A, thus the first factor becomes S * n 

 2 n sin A 



Now sin nA = sin{A-f(w — 1)A} = sin (n — l)Acos A -f cos — 1) A 



sin A, and sin (n— 1)A cos A=sin (n— 2) A cos 3 A + cos (n— 2) A cos A 



sin A ; and, similarly, 



sin (n— 2)Acos 2 A=sin (n— 3) A cos 3 A + cos (n— 3) A cos 2 A sin A. 



sin 3 A cos (w_3) A= sin 2 A cos (/i * 2) A + cos 2 A cos u_3) A sin A. 

 sin 2A cos u-2) A= sin A cos (re_1) A + cos A cos" -2 A sin A. 



Thus 



sinwA=sin A{cos(?i— l)A-f-cos(w— 2)Acos A-f cos(n— 3) A cos 2 A . . . 



cos 2 A cos w-3 A + cos A cos" -2 A + cos" -1 A} . 



The number of terms on the right hand side of the equation is n, and 

 the numerical sum (regardless of sign) of the terms in the bracket is 

 obviously less than n, thus the numerical value of sin nA is less than 

 the numerical value of n sin A ; and since cos A is numerically less 

 than unity, the other factor is also a proper fraction. 



10. The following conditions will show without proof when the 

 first and the second factors respectively will have the positive or 

 negative sign : — 



The first factor is positive when nb lies between 27r(2N) and 

 2tt(2N + 1), and b lies between 2tt(2N') and 2 7 r(2N , + 1), or when 

 nb lies between 2^(2^ + 1) and 27r(2N + 2), and b lies between 

 2*-(2N / + l) and 2rr(2W + 2). 



