Periodicities when the Periods are unknown. 367 



we have 



clhi __sin^_2cosa3_|_2 cosa? 

 dx 2 x x 2 Sx 2 ' 



_ _sin x 4 cos «__2_ c0 



Thus when aj=0, is negative. 



eta? 2 



Therefore when x=0 u has a maximum value. Thus we see that 

 as the angle nb changes from to 2 77-, the factor diminishes from the 

 limiting value unity to zero. 



We shall next see when we shall have a minimum value of the 

 factor when nb changes from 2-7T to 4<7r. 



Put -^-=7r-{-x, and consequently n= — ^ — ; thus the first factor 



becomes- 



sm(7r + a3)_2 v _ 2 



sin- (7r + aj;sm— (77- + a;) sm- 



b a 2 2 



— 2 sin a; 



Let _ , remove constant factors, 



(tt + x) sin- 



, , — sin x 

 thus u= , 



TT + X 



then 



du_ — (7r -f x) cos x + sin x 

 dx (7r + xy 2 



d 2 u _ sin x , 2 cos x 2 sin a; 

 cfo; 2 7r-\-x (tt + x) 2 (7r-\-x) % 



du . 77°'5 



If we put — =0 we obtain tan#=7r + a?, therefore x= — — . 



dx 57 '3 



Therefore when x=0, 



d 2 u -97630 ^2(-21644) 2(-97630) , , , . 



— -= +-r -— — z_=nlus something; 



da» 4-493 (4-493) 2 (-493) 3 



77°*5 



therefore when a3 = -— — , it has a minimum value. 

 57 c -3 



Thus when nb=2^+—\ we have a minimum value, and similarly 

 vol. xxxvi. 2 c 



