40  Practical  Method  of  Harmonic  Analysis. 
This  curve  is  the  parabola 
IT 
whose  axis  is  w  =  7r/2  and  vertex  #=7r/2,  ?/  =  ?w17r/4;  and 
the  harmonic  expression  for  the  wave  of  which  it  is  the  type 
is  obtained  by  making  ft  — co  and  a  =  0  in  the  expression  in 
(c),  and  is 
8m-i  f   .       ,       sin  3wt   ,  sin  5cot      0     ~) 
-jT^smot  +— 33— +— 53— +&c.|. 
13.  To  find  the  harmonic  expression  for  the  complete 
periodic  function  whose  graph  for  one  period  is  made  up  of 
the  sides  of  two  equal  and  similar  triangles  ABC  and  A'BC 
so  placed  that  A'  and  C  lie  in  AB  and  CB  produced  respec- 
tively. Take  A  as  origin,  then  the  abscissae  of  B  and  A/  will 
be  7r  and  2ir  respectively  ;  and  let  the  abscissa  of  C=/x5 
hence  that  of  C/  =  27r  — ^. 
By  geometrical  construction  the  different  components 
of  this  wave  can  be  easily  obtained  if  we  remember 
formula  I.  §  1. 
Thus,  to  get  the  2nd  component  we  cut  the  wave  in  four 
portions  by  ordinates  at  7r/2}  it,  3*7t/2,  2tt,  invert  the  second 
and  fourth  portions,  superpose  them  and  the  third  portion  on 
the  first,  add  the  corresponding  ordinates,  divide  each  sum 
by  four,  and  the  plot  of  the  results  will  be  a  half-wave,  which 
gives  the  2nd  component. 
It  will  be  found  for  the  wave  under  consideration  that  all 
the  components  are,  in  general,  trapeziums  of  the  type  treated 
in  §  12(a) ;  and  if  the  trapezium  which  is  the  rth  component 
be  specified  as  in  §  12(a)  by  mr  and  fir  measured  on  the 
original  scale  of  abscissae,  it  will  be  found  that 
tan  A  +  tan  B 
»,=  ±  2r 
(i.e.,  that   2r  times  the  function   of   its  vertex  is  equal  to 
+  the  function  of  the  vertex  of  the  triangle) ;  and  that 
sin  Tfj,r  =  +  sin  r/x, 
the  same  signs  being  taken  together. 
It  is  to  be  noted  that  as  the  base  of  the  rth  component  is 
ir/r,  the  altitudes  of  its  isosceles  elements  are  each 
2r1L       ir     r> 
