Practical  Method  of  Harmonic  Analysis.  25 
have  in  every  case  a  finite  intensity  of  source,,  or  of  forcive, 
except  in  §  80  where  we  have  supposed  b  infinitely  small,  in 
comparison  with  A,  we  avoid  the  infinity  shown  in  column  G : 
and  can,  by  great  labour,  calculate  a  table  of  mitigated 
numbers,  rising  to  a  very  large  maximum  at  ty=  ±35°  16'; 
but  not  to  infinity  ;  and  so  arrive  mathematically  at  an 
expression  for  the  very  high  waves  seen  on  the  two  bounding 
lines  of  the  wave-disturbance,  inclined  at  19°  28'  to  the  mid- 
wake.  But  it  is  interesting  to  remember  that  we  see  in 
reality  a  considerable  number  of  white-capped  waves  (would- 
be  infinities)  before  the  well-known  large  glassy  waves  which 
form  so  interesting  a  feature  of  the  wave-disturbances. 
§§  80-95  of  the  present  paper  is  merely  a  working  out 
of  the  simple  problem  of  purely  gravitational  waves  with  no 
surface-tension  on  the  principle  given  by  Eayleigh  *  in  1883 
for  the  much  more  complex  problem  of  capillary  waves  in 
front,  in  which  surface-tension  is  the  chief  constituent  of  the 
f orcive,  and  waves  in  the  rear,  in  which  the  chief  constituent 
of  the  forcive  is  gravitational. 
In  all  the  work  arithmetical,  algebraic,  graphic  of  §§  32-95 
above,  I  have  had  much  valuable  assistance  from  Mr.  J. 
de  Graaff  Hunter  ;  who  has  just  now  been  appointed  to  a 
post  in  the  National  Physical  Laboratory. 
II.    On    an    Expeditious     Practical    Method    of    Harmonic 
Analysis  f.      By  Thomas  R.  Lyle,  M.A.,  Professor  of 
Natural  Philosophy  in  the  University  of  Melbourne. 
[Plate  L] 
l.T^OURIER  has  shown  that  if  any  function  fit)  (=ysay) 
X       of  a  variable  t  be  such  that 
/(0=/(*  +  t)=/(*  +  2t)=&c, 
where  t  is  a  constant,  that  is,  if  f(t)  be  periodic  in  t,  of 
period  r,  then  f(f)  can  be  expressed  as  the  sum  of  a  constant 
and  a  series  of  terms  called  harmonics,  each  of  the  form 
ap  sin p{a>t- Op), 
where  p  has  the  values  1,  2,  3,  4,  &c,  and 
O)  =  27t/t. 
*  Proc.   Lond.  Math.  Soc.  xv.  pp.   69-78,  1883;    reprinted  in   Lord 
Rayleigh's  '  Scientitic  Papers,'  vol.  ii.  pp.  25&  267. 
t  Appendix  to  the  papers :  " Preliminary  Account  of  a  Wave-Tracer 
and  Analyser,"  Phil.  Mag-.  Nov.  L908,  and  "Investigation  of  the 
Variations  of  Magnetic  Hysteresis  with  Frequency,  Phil.  Mag", 
Jan.  1905.  Reprinted  from  a  separate  copy,  communicated  by  the 
Author,  of  the  'Proceedings'  of  the  Royal  Society  of  Victoria,  Vol. 
(n.  s.)  pt.  2,  Feh.  1905. 
