34  Prof.  T.  R.  Lyle  on  an  Expeditious 
number  is  specially  suitable,  as  it  enables  us  to  determine 
directly  C\,  C2,  C3,  C4,  and  C6.  To  determine  C5,  replotting 
will  have  to  be  resorted  to  if  the  full  wave-trace  be  not 
available. 
At  the  top  of  Table  III.  are  written  the  24  given  ordinates 
under  their  corresponding  abscissae.  From  these  ordinates 
the  constant  term  of  f(t)  has  been  removed.  This  can  be 
done  by  aid  of  the  formula 
/(0+/('  +  T/n)+/(«  +  2r/n)+  ....  +/(*+ii=Ir/») 
—n  [a0  +  an  sinrc  {cot  —  6n)  +a2n  sin  2n(cot  —  02n) 
+  fl3wsin3n(^~^3J+&c],     ....     (III.) 
which  can  be  easily  established  by  the  method  used  in  §  2. 
From  this  formula  we  see  that  the  mean  of  n  e.s.  ordinates 
embracing  one  period  of  a  periodic  function  is  equal  to  its 
constant  term,  if  its  wth,  2nth,  &e.  harmonics  are  neglected. 
Returning  to  Table  III.,  we  add  the  second  twelve  ordi- 
nates with  their  signs  changed  to  the  first  twelve,  in  order, 
and  obtain  12  e.s.  ordinates  of  20^  i.e.  of  2  [H1  +  H3  +  H5  -j-  &c] . 
(See  equation  I.,  §  2.) 
Subtracting  these  from  twice  the  given  ordinates,  those  of 
2  [H2  +  H4+  H6  +  &c]  are  left,  and  the  remainder  of  the  work 
proceeds  as  in  Table  I. 
2  [H2  +  H4  +  H6+  &c.]  could  be  obtained  directly  from  the 
24  given  ordinates  by  adding  the  second  12  to  the  first  12  of 
them,  in  order.     (See  formula  III.,  §  9.) 
The  amplitudes  and  phases  of  the  different  harmonics  were 
determined  as  in  Table  L,  but  the  figures  necessary  iu  their 
calculation  are  not  given. 
The  following  are  interesting  applications  of  the  above 
method  to  more  general  harmonic  analysis. 
To  obtain  the  harmonic  expression,  for  the  odd  periodic 
function  whose  graph  for  half  a  period  is  the  sides  of  an 
isosceles  triangle  of  altitude  h.      (See  fig.  1.) 
Taking  0  and  ir  as  the  abscissae  of  the  extremities  of  the 
base,  relative  values  of  any  number  of  e.s.  ordinates  can  be 
written  down,  and  any  component  at  once  obtained.  Thus, 
30  e.s.  ordinates  would  be  0,  1,  2,  3,  .  .  14,  lo,  14,  ...  2,  1, 
and  these  correspond  to  an  altitude  15. 
It  will  be  found  that  all  the  components  (i.  e.  3rd,  5th,  &c. 
in  this  case)  are  the  sides  of  isosceles  triangles  passing 
through  the  origin,  and  that  the  altitudes  are 
— /?/32,    h/52,    —  hj I2,  &c.  respectively.     (See  fig.  1.) 
(The  same  can  be  quickly  arrived  at  geometrically.) 
