Practical  Method  of  Harmonic  Analysis.  39 
When   fi  =  120°   the   above    expression    for    the    triangle 
reduces  to 
3v  7  ^  J  sm(a)£-/3)-  — — 52  + — ^2 --&C.J- 
4A    (        n    ,      eos9o)^  ,   cosl5&)£       „      ) 
+  3-3  {  008SW+  -gr-  +  -jf-  +&c.  | , 
where  .       _  1 
tan  0  = 
3v/3 
12(c).  Wave-form  a  polygon  with  n  vertices  per  half-wave 
and  such  that  the  functions  of  its  vertices  are  all  equal  and 
also  the  projections  of  its  sides  on  the  axis  of  x. 
Let 
q=  mx— m2  —  m2  —  m-6  —  m3  —  m4  =  &c.  =  mB  +  w1, 
and  let  the  abscissae  of  its  vertices  be 
a,   a  +  irjn,    a  +  27rn,  .  .  .  .  a  +  (/i  —  l^ir/n  ; 
then  by  §  11  the  expression  for  the  wave  is 
^S.N(«f  +  w/2-[a  +  rw/n]), 
where  t1  has  all  values  from  0  to  n—1. 
Substituting  for  the  N  functions  their  equivalent  harmonic 
series,  summing  the  terms  that  have  the  same  arguments  and 
remembering  that  nq  =  2mi,  the  expression  for  the  polygonal 
wave  under  consideration  becomes 
4m , 
nir 
\ sin(W  —  a  +  7r/2w)  H ^—  sin  3  (a>/ —  a  +  7r/2  u) 
suit-  3-  sin  — 
In  Z/i 
H ?—  sin5(a>f— a  +  7r/2w)H-&c.  /■ . 
52  sin-— - 
zn 
12(d).  If  in  example  (c)  ra  become  infinite,  the  polygon  In- 
comes a  smooth  curve  satisfying  the  following  conditions  : — 
*2  =  const., 
dil  i^i  i 
;    =mi  when  #  =  0,  ami  =-Wi  when  #  =  7r, 
rto? 
y  =  0  when  a?=0  and  when  r  — 7r. 
