Practical  Method  of  Harmonic  Analysis.  37 
Hence  nx  —  n2  =  ml  —  m2, 
n2  —  nz  =  m2—m3) 
n3  —  n3'—  2M, 
%  — %  =  m4  — m5. 
Thus,  i£  we  call  m1  —  m2  the  function  of  the  vertex  #i2,  we 
see  that  by  addition  of  /  to  the  given  wave  a  new  vertex 
is  introduced  whose  function  is  equal  to  that  of  1(  =  2M), 
while  the  functions  of  all  the  other  vertices  are  unchanged. 
It  is  easy  to  see  that  the  function  of  the  vertex  of  —  /  is 
—  2M,  so  that  if  /  be  subtracted  from  the  given  wave  a  new 
vertex  is  introduced  whose  function  is  —  2M. 
If  the  abscissa  X  of  the  vertex  of  /  correspond  with  that 
of  one  of  the  vertices  of  the  given  polygon  x23>  say,  then  no 
new  vertex  will  be  introduced  by  the  addition  (or  subtrac- 
tion) of  /,  and 
ni  —  n2  =  nil  —  m2> 
n2  —  ?z3  =  m2  —  ?»3  +  2M, 
nz  —  rc4  =  m3  —  ml} 
n±—n5  —  m±  —  mh. 
If,  in  addition,  2M=m3-m2,  then  n2  —  n^  —  D  and  the 
vertex  or  break  at  x2z  is  removed.  Thus,  by  subtracting 
from  the  polygonal  wave  an  isosceles  wave  whose  vertex  has 
the  same  function  and  abscissa  as  a  vertex  of  the  given 
polygon,  this  vertex  of  the  polygonal  wave  is  removed,  while 
the  functions  of  its  remaining  vertices  are  unchanged.  To 
remove  each  vertex  therefore  a  definite  isosceles  wave  is 
required,  and  since,  when  all  vertices  are  removed  the  axis 
of  abscissae  or  y  =  0  remains,  we  see  that  the  sum  of  the 
several  isosceles  waves  required  to  extinguish  the  given  wave 
is  equal  to  the  latter. 
In  the  general  case,  therefore^  of  a  polygonal   wave  with 
n  vertices,  the  vertex  m.  m  ,,,  a:      L1   will  be  removed  by 
subtracting  the  isosceles  wave  I=]LN(cot  —  oi),  where 
mr— mrH  =  2M  =  ih/ir, 
and  the  complete  wave  will  be  fully  represented  by 
of  the  n  isosceles  functions  given  bv 
J   (vn    _.,,,         ^  AV  ,.*/__  ..  J    —  -^  1 
the  sum 
■77-  r=n  +  \ 
4      ,tl     Omr"mH-l)N(«'"'*rkr+l 
remembering  thai  »&n+1=  — iwlt 
