32  Prof.  T.  R.  Lyle  on  an  Expeditious 
From  a  given  number  of  e.s.  ordinates  only  an  approxi- 
mate analysis  can  be  obtained,  more  approximate,  of  course, 
as  the  number  of  ordinates  is  greater.  When,  however,  each 
individual  ordinate  has  been  obtained  with  the  accuracy  of 
which  the  galvanometer  and  scale  method  is  susceptible,  the 
analysis  obtained  from  fifteen  such  ordinates  is  much  more 
reliable,  as  far  as  the  harmonics  up  to  the  9th  are  concerned, 
than  that  determined  from  any  photographic  trace. 
I  will  therefore  illustrate  the  method  by  applying  it  in  full 
detail  to  the  analysis  of  the  wave  whose  15  e.s.  ordinates  are 
given  in  row  5  of  Table  I.  (PL  I.).  Every  figure  necessary 
in  the  calculation  will  be  given. 
The  first  row  of  figures  in  Table  I.  are  the  abscissae  <r0,  #ls 
&c,  to  which  the  given  ordinates  correspond.  Space  for 
three  rows  of  figures  is  left,  and  then  the  15  given  ordinates 
are  written  down.  These  are  divided  into  three  sets  of  five 
each,  and  the  numbers  of  the  middle  set  are  subtracted  in 
order  from  the  sums  of  first  and  last  set,  giving  five  numbers 
which  are  the  corresponding  ordinates  of  3C3.  Space  for 
two  or  more  rows  is  left,  and  the  given  ordinates  are  now 
written  down  as  in  the  table,  in  two  rows  of  six  each  and 
one  row  of  three,  in  order.  The  columns  formed  are  added 
and  the  last  three  of  the  sums  are  subtracted  from  the  first 
three,  giving  three  ordinates  of  5C5.  The  first  of  these  minus 
the  second,  plus  the  third,  gives  one  ordinate  of  15C15,  whose 
other  ordinates  are  got  by  alternating  the  sign.  Subtracting 
5015  from  5C5  we  obtain  5H5.  Having  obtained  C15  we  now 
subtract  3C15  from  3C3  and  obtain  3  (H3-j-H9). 
Above  the  given  ordinates  write  those  of  C3  with  signs 
changed  (row  4),  and  above  these  write  those  of  H5  with 
signs  changed  (row  3).  Add  rows  3,  4,  and  5  to  get  row  2, 
in  which  are  the  ordinates  of  Hx  -f  H7  +  Hn  &c.  Neglecting 
H7,  Hn,  &c,  as  is  done  in  the  analysis  in  Table  I.,  we  may 
consider  the  figures  in  row  2  as  the  ordinates  of  Hj,  and 
neglecting  H9  we  may  consider  the  figures  in  row  11  as  the 
ordinates  of  3H3. 
The  first  15  numbers  under  Amp.  Hi  are  the  quarter  squares 
of  the  ordinates  of  Hi.  Twice  the  sum  of  these  is  divided  by 
15,  the  number  of  ordinates,  and  the  quotient  is  found  to  be 
the  quarter  square  of  987.  Hence  h1}  the  amplitude  of  H1; 
is  987.     Similarly  for  the  amplitudes  of  H3  and  H5. 
Under  the  heading  "  Phase  of  H^  in  the  first  column 
under  sines,  are  the  quotients  got  by  dividing  the  first  four 
ordinates  on  the  rising  side  of  Hi  and  the  last  four  on  the 
falling  side  of  H1  by  hi ;  in  the  second  column  under  angles 
are  the  corresponding  angles,  and  in   the  third  column  are 
