Practical  Method  of  Harmonic  Analysis.  31 
respectively,  and  if  A1?  a  be  the  amplitude  and  phase  of  H,, 
or  in  other  words  if 
Hi  =  71!  sin  (cot  —  a), 
then  any  of  the  equations 
sin  (#0  —  a)  =  ajhi, 
sin  (x1  —  ct)  =  al/h[, 
sin  {x^  —  oL)  —  a^\h^  &c. 
would  determine  a,  provided  the  ordinates  a0,  a1?  «2,  &c.  are 
exactly  those  of  a  sine  wave. 
In  practice,  however,  small  upper  harmonics  will  invariably 
be  left  in  «0,  al5  a2  &c.  [it  may  not  have  been  thought  worth 
while  to  remove  H7],  and  though  their  amplitudes  may  be 
negligibly  small,  yet  they  might  cause  considerable  error  in 
the  value  of  a  when  determined  from  only  one  of  the  above 
equations.  Hence  it  is  advisable  to  obtain  four  values  of  a 
from  the  first  four  ordinates  on  the  rising  side  of  the  wave 
and  four  from  the  last  four  ordinates  on  the  falling  side,  and 
take  the  mean  of  the  eight.  In  this  way  we  can  to  a  great 
extent  eliminate  any  error  that  might  arise  through  an  har- 
monic even  as  low  as  the  seventh  not  having  been  removed. 
In  a  similar  way  the  phases  of  H3,  H5,  &c.  can  be  deter- 
mined, but  it  must  be  remembered  that  if,  for  instance, 
H3  =  A3  sin  3  (cot— ft), 
and  if  bQ,  61?  .  .  .  .  54  are  the  ordinates  of  H3  corresponding 
to  the  abscissae  #0,  #l5  ....  #4,  then 
sin  3  (^o—/5)=60/^3,  &c.  ; 
similarly,  if 
H5  =  /i5  sin  5  (a>£  — 7), 
with  ordinates  c0,  c1}  c2,  then 
sin5  0o-Y)  =  co/7i5- 
7.  The  wave  to  be  analysed  may  be  given  in  either  of  two 
ways.  We  may  have  the  complete  trace  of  it  obtained  by 
the  author's  wave-tracer  by  the  photographic  method,  or  by 
any  form  of  oscillograph  that  gives  a  trace  of  the  wave  form: 
or  we  may  have  the  values  of  a  definite  number  only  of 
ordinates  per  half-wave,  such  as  would  be  obtained  by  the 
author's  wave-tracer  by  the  galvanometer  and  scale  method. 
From  the  wave-trace  the  complete  harmonic  expression 
can  theoretically  be  obtained,  but  the  impossibility  ot  accu- 
rately measuring  on  the  photograph,  without  elaborate 
apparatus,  the  different  ordinates  required  leads  to  great 
inaccuracy  in  the  result. 
