38  Prof.  T.  R.  Lyle  on  an  Expeditious 
The  following  are  examples  of  the  preceding  method  : — 
12(a).  Wave-form  a  trapezium  with  equal  base  angles. 
This  is  the  sum  of  two  equal  isosceles  waves. 
Take  the  left  extremity  of  the  base  of  the  trapezium  as 
the  origin  of  the  abscissas,  and  let  it  be  specified  by 
m1 = m,    m2  =  0,   ?n3  =  —  m,    «i2 = ^,    a2z = it  —  fx, 
so  that  its  altitude  t  =  /j,m. 
By  §  11  the  expression  for  the  wave  is 
™  [N(^-/.  +  7T/2)  +  N(^  +  ^-7T/2)] 
2rar.    ,              ,    \a\      sin  3  (W— At  +  tt/2)       „ 
=  ■ — •    sin  (art—  u  +  w/2) — ^ +ac. 
7T    L  6" 
,    •    /    ..             /on      sin3(&>£  +  jU  — 7r/2)       p     1 
+  Sin(ft)^  +  /Lt  — 7T/2)—  i — L^.  +  &c< 
4tf  r  .         .       ^  ,  sin  3|it  sin  3&>£ 
—    sinu  sin  ©£  H ^r^— 
(Mir  L 
jhtL      ^  '  32 
,  sin  5a  sin  5o>£      „    n 
+ ^ +&c.J, 
which  is  Fourier's  expansion  for  a  wave  of  this  form. 
12  (b).  Wave-form  a  triangle.  This  is  the  difference  of 
two  isosceles  waves  when  the  vertex  of  one  lies  on  a  side 
of  the  other. 
Take  the  left  extremity  of  the  base  of  the  triangle  as  the 
origin  of  abscissas,  and  let  it  be  specified  by 
m1=m,    m2=—n,    md=z —m,    ai2  =  fi,    a23  =  7r, 
so   that   its   altitude    h  =  fim={nr—fi)ni    and   m,n   are    the 
tangents  of  its  base  angles. 
By  §11  its  expansion  in  isosceles  functions  is 
j  [(m  +  n)  ~R(cot  -ft  +  tt/2)  +  (m  -  n)  N(W  —  tt/2)] 
=  — [N(»/-/i  +  7r/2)  +  N(orf-w/2)] 
+  ™  [Br(ffl^-/»  +  w/2)-N(«rf-w/2)], 
which  is  o 
.                                              sin-^-sin3(&)^— ju/2) 
=  —     sin  -  sin  (cot—/j/2)  + ^ •"  &c- 
3a 
.     _  cos-^-cos3(a>£  — a/2) 
H i  cos"-cos(a>/  — a/2)  + ^ -f&c. 
where  mji  —  n  [ir  —  //,)  =  7? . 
