Practical  Method  of  Harmonic  Analysis.  33 
the  eight  values  of  12°  — a  deduced.  The  mean  of  these 
2°  2'  when  subtracted  from  12°  gives  the  crossing-point  or 
phase  of  Hx  as  9°  58'.  Similarly  for  the  phases  of  H3  and 
H5.  It  will  be  noticed  that  at  the  crossing-point  determined 
for  H3,  H3  crosses  down,  which  is  expressed  analytically  by 
writing  its  amplitude  negative. 
8.  It  will  be  noticed  in  the  determination  of  the  phase  of 
Hx  in  Table  I.,  that  the  eight  values  of  12°  —  a  differ  consider- 
ably from  each  other,  indicating  the  presence  in  what  we 
there  take  for  H^  of  a  considerable  upper  harmonic,  pro- 
bably H7.  In  order  to  determine  H7,  fourteen  e.s.  ordinates 
of  the  half- wave  are  required.  If  the  wave-trace  were  given 
these  could  be  measured  off  from  it,  but  if,  as  in  the  case  we 
are  considering,  only  15  original  ordinates  are  given,  it  is 
necessary  to  plot  the  15  ordinates  of  Hx  +  Hj  obtained  in 
Table  I.,  and  from  the  smooth  curve  drawn  through  them  to 
measure  off  14  e.s.  ordinates.  This  has  been  done  and  the 
values  obtained  are  given  in  row  4,  Table  II.,  as  well  as  the 
calculation  necessary  for  the  determination  of  H7  and  its 
elimination  from  Hx  +  H7. 
What  is  called  the  amplitude  of  Hx  in  Table  I.  is  really 
\/2  R.M.S.  (Hj  +  H7) .  To  get  amp.  E±  it  is  better  to  remove 
the  effect  of  H7  by  treating  it  as  a  correction,  thus  avoiding 
error  that  might  be  introduced  in  the  plotting.  This  is  easily 
done,  since 
M.S(H1  +  H7;=^  +  ^; 
hence       \  (corrected)  =  \/Amp.  (Hi -I-  II 7)2  —  h72, 
=  a//*!2  (uncorrected)  —  /^2 
In  Table  II.  the  corrected  crossing-point  of  B^  is  deter- 
mined, and  it  is  seen  to  differ  in  phase  only  by  2  minutes 
from  the  value  obtained  in  Table  I. 
The  differences  between  the  four  values  of  3  (24°— 0)  when 
determining  the  crossing-point  of  H3  in  Table  I.  point  to  the 
presence  of  a  ninth  harmonic,  which  exists  as  a  third  com- 
ponent in  H3  +  H9.  H9  can,  if  desired,  be  determined  by 
plotting  the  five  ordinates  obtained  in  Table  L,  measuring 
off  from  the  curve  six  e.s.  ordinates,  and  proceeding  as 
before.     It  will  be  found  that 
H9  =  3sin9(a>/f-13°). 
9.  In  Table  III.  is  given  most  of  the  work  required  for  the 
determination  of  the  first  six  harmonics  of  a  complete  wave 
that  contains  harmonics  both  of  odd  and  even  orders. 
Twenty-four  e.s.  ordinates  of  the  full  wave  are  taken.      This 
Phil.  Mag.  S.  6.  VoJ.  11.  No.  61.  Jan,  1906,  V 
