28  Prof.  T.  R.  Lyle  on  an  Expeditio 
'US 
and  if  f(t)  is  an   odd   periodic   function  as  in  the   case   of 
alternate-current  waves  which  we  are  now  considerino- 
Lo' 
9  W  =  -gQe  +  Tr)  =g(x  +  2tt). 
Substituting  g(x)  for  f(f)  in  equation  II.,  it  becomes 
nG»=g(*)—gQe  +  'irln}+g(a:  +  2irlri)  - 
■\-g(js  +  n  —  lirji{), 
from  which  we  conclude  that,  if 
go,  gi,  y* yn-i 
be  n  equi-spaced  ordinates  that  exactly  include  half  the  wave, 
i.  e.  ordinates  corresponding  to  the  abscissas 
x,  x  +  tt/ii,    x  +  2-njn, x  +  n  —  lirjn 
respectively,  and  called  e.s.  ordinates  in  the  sequel  ;  and  if 
N„,  N„  N2, N„_5 
be  the  ordinates  of  the  ?zth  component  Gn  whose  abscissae 
are  the  same  as  those  of 
go,gi,g2, gn-x 
respectively,  then 
yo— yi+y«-  •  •  •  +^_1=wN0=^nN1=wN2=     =nNn_i 
when  n  is  an  odd  number,  and 
yo-gi+g2- -y»-i=o 
when  n  is  an  even  number,  as  we  are  now  considering  odd 
periodic  functions  only. 
Thus  from  n  e.s.  ordinates  of  the  original  half-wave  we 
obtain  only  one  ordinate  per  half-wave  of  0  ,  so  that  in  order 
to  obtain  m  e.s.  ordinates  per  half -wave  of  (Jn  it  is  necessary 
to  have  mn  e.s.  ordinates  of  the  original  half-wave. 
For  instance,  to  obtain  3  e.s.  ordinates  of  Gn  we  must 
measure  3n  e.s.  ordinates  of  g(x).     Let  these  be 
go,  y»  g2,  gs, g3n_v 
and  let  the  corresponding  ordinates  of  Gn  be 
N0,  N„  N2I  N3 Nfc_y 
then 
go— yi+ye—  •  •  •  +gSn.3  =  n^0=-nN3  =  n^6  =      =nN3n_8, 
Vi-g^gi-  •  •  •  +^-2=nNi=-wN4=«N7=  =nN^,2s 
g-2-go+gs=  •  .  ■  +yfc_1=nN,=  — nNB=iiNg=     =«N3/l_1. 
