30  Prof.  T.  R.  Lyle  on  art  Expeditious 
component  of  C5,  of  which  we  have  3  e.s.  ordinates  ?/0->  uu  u2) 
its  three  corresponding  ordinates  are  i0,  —iQ}  i0}  where 
3i0  =  u0  —  ux-{-u2. 
Hence  H5  will  be  completely  given  by 
c0,    c1}    c2,     where 
cQ  =  Uq  —  hi   cr= «!  +  »(,»    c2=U2—i0. 
H15  can  now  be  taken  from  C3,  thus 
r0~?0.>     -1  +  ^0?     Z2       l03     ^3+2C5     ~4 —  20 
are  the  5  e.s.  ordinates  of  H3  +  H9. 
In  order  to  determine  H3  and  H9  it  will  now  be  necessary 
to  plot  the  5  ordinates  of  H3  +  H9,  measure  off  6  e.s.  ordinates 
from  the  smooth  curve  drawn  through  them,  and  from  these 
determine  their  first  component,  that  is  2  e.s.  ordinates 
of  H9.  These  will  completely  determine  H9  if  H27  &c.  be 
neglected,  and  by  subtracting  them  from  the  corresponding 
ordinates  of  H3  +  H9  6  e.s.  ordinates  of  H3  are  obtained. 
If  H7  cannot  be  neglected  it  will  be  necessary  (if  the 
original  wave-trace  is  not  available)  to  plot  the  15  ordinates 
of  Hi  +  H7  obtained  above,  and  from  the  smooth  curve 
drawn  through  them  to  measure  off  14  e.s.  ordinates.  From 
these,  2  e.s.  ordinates  of  the  half-wave  of  H7,  which  deter- 
mine H7,  can  be  obtained.  By  subtracting  these  from  the 
corresponding  ones  of  Hx  +  H^  14  corrected  ordinates  of  Hx 
are  obtained. 
6.  It  now  remains  to  determine  the  amplitudes  and  phases 
of  the  harmonics  of  d  from  their  ordinates  which  we  have 
obtained.     It  is  easy  to  show  that 
||sin204-sin2('6'  +  ^+sin2^+^) 
+  sin2  (fl+^I  ^)}"=t 
from  which  we  conclude  that  the  square  root  of  twice  the 
mean  of  the  squares  of  n  e.s.  ordinates  of  half  a  sine  wave  is 
equal  to  its  amplitude. 
Hence,  with  the  help  of  a  table  of  squares  or  of  the  quarter 
squares  given  in  most  sets  of  tables',  the  amplitudes  of 
Hi,  H3,   &c.  can  be  quickly  determined. 
[The  rule  that  the  amplitude  is  equal  to  irj2  x  mean  of 
the  ordinates  is  only  sufficiently  accurate  when  a  large 
number  of  ordinates  is  taken.] 
If  a0i  av  a2,  .  .  .  .  au  be  the  ordinates  we  have  found  for 
H2  corresponding  to  the  angular  abscissae  *r0,  xx,  x2,  ....  xu, 
