36  Prof.  T.  R.  Lyle  on  an  Expeditious 
But  C!  =  0  when  cot  =  0,  therefore  61  =  0  ;  and  C^h  when 
cot  —  irj'l  ;  therefore 
/i-a1[l  +  l/32  +  l/5M-l/72-f&c.]=a17r2/8. 
Hence  a1  =  8h/7T2J  and  the  Fourier  series  required  is 
~       87i  r  .       ,      sin  Scot      sin  5cot      D    ~1 
Cl=^2LSm §2~+   ~~ 52 &C,J 
=  JiN  (cot)  say. 
If  the  left  extremity  of  the  base  were  at  a  distance  a  from 
the  origin  instead  of  coinciding  with  it,  then 
G^hNicot-a). 
Let  us  call  the  function  hN(cot  —  a)  an  isosceles  function, 
and  the  series  of  isosceles  triangles  which  is  the  graph  of 
his  (cot— a)  an  isosceles  wave  specified  by  h  its  altitude,  and 
a  its  phase. 
11.  Any  wave  containing  only  odd  harmonics  whose  form 
is  polygonal,  with  n  vertices  per  half-wave,  can  be  resolved 
into  n  isosceles  waves  of  the  same  period,  and  hence  can  be 
analytically  represented  by  a  sum  of  n  isosceles  functions. 
A  vertex  may  or  may  not  occur  where  the  wave  crosses 
the  axis  of  abscissae.  In  the  latter  case  the  base  angles  of 
the  polygon  will  be  equal. 
For  the  sake  of  definiteness  let  us  consider  the  case  when 
the  polygon  has  4  vertices  per  half-wave,  and  let  it  be 
specified  by  m1?  w2,  wi3,  m4,  mb  (m5= — ?%),  the  tangents  of 
the  angles  its  sides,  taken  in  the  positive  direction,  make 
with  the  axis  of  x,  and  by  the  abscissae  xi2,  #23,  #34*  ^45,  of 
its  vertices. 
In  the  first  place  let  us  determine  the  form  of  the  wave 
got  by  adding  to  the  above  the  isosceles  wave  I  =  fiN(cot  —  a) 
specified  in  the  above  manner  by  M,  —  M,  and  X,  so  that 
h  =  M.7r/2  and  a  +  7r/2  =  X.  In  general  the  new  wave  will 
have  5  vertices,  the  abscissa  of  the  one  introduced  being  X, 
while  the  abscissae  of  the  others  are  unchanged,  and  if  X  lie 
say  between  x2z  and  <r34,  and  the  tangents  of  the  slopes  of  the 
sides  of  the  new  polygon  be  n1}  n2i  n3,  n3',  ti4,  —  n1}  then, 
remembering  that  the  equations  of  the  different  sides  are 
of  the  form 
yz=zmx-\-K3 
we  see  that 
nx = mx  +  M,     n2  =  m2i  M,     ?i3 = m5  +  M, 
n3'  =  m3 — M,     nA  =  m±  —  M. 
