OF OBSERVATIONS OF COMPLEX PERIODICAL PHENOMENA. 
375 
or =X a m sin tmz 
r«( /n i n ,® — ^ \ ( n n 1 riW s 2 i 2 f /_ n 2 s— \ 
+ \ L 2[^[ 0 +2 COt ~2~ Z )~^\~2~2 + 2)\-^\^{ 0+ 2 COt ~2~ Z ) 
, / n 2 . n 1 w 3 n 2 
or=£ a m sin tmz, 
according as neither (s+£);s nor (s — tf )2 is 0 or a multiple of 27 t; as (s-\-t)z is not, 
but (s— t)z is 0 or a multiple of 2-n- ; as ( s-\-t)z is, but (s— t)z is not 0 or a multiple of 
2?r; or as both (s-\-t)z and ( s—t)z are multiples of 2tt. 
2 m=n—\ 2 m=n-l 
- X oc m > sin tmz—-- X a m sin tmz 
71 7ft = 0 U 771 = 0 
+[f {&( cot S ~Y Z ~ cot | — S ~rfes(n cot S ~^z—n cot 
/ 1 
1 \ 
A • 0 ® t 
• 0 S+t ) 
v sm 2 - — z 
Sin 2 -rr~Z' 
2 
2 
< 
or= : 
+[l{^(- cot ^)-^}-x{^(“ wcot ^)+^(x +^“777+7)}]’ 
> (27) 
2 m=n— 1 
or=- X a m sin a sin tmz 
-[!{& (cot S -^z)j -~i-\p s n cot +&( - 1 w 2 - 1 + ) j j, 
2 m=n-l 
or=^X a m sin tmz. 
according as neither ( s-{-t)z nor ( s—t)z is 0 or a multiple of 27 r; as ( s-\-t)z is not, 
but (s—t)z is 0 or a multiple of 27r; as ( s-\-t)z is, but (s— t)z is not 0 or a multiple of 
2? r; or as both (s +#)2 and ( s—t)z are multiples of 27 r. 
And writing b t and b f respectively for the coefficients of i and i 2 in (27), and 
transposing, 
q-Qt-b, t i-bfi 2 (28) 
