Constitution  of  Natural  Radkition.  125 
The  difference  is  the  aggregate  o£  PQ.  Now  we  suppose  that 
t  is  so  related  to  the  greatest  hi  that  all  the  arcs  PQ  are  small 
fractions  o£  the  quadrant,  and  the  question  before  us  is  the 
amount  of  the  difference  between  the  resultants  of  the  OP's 
and  the  OQ's,  i.  e.  of  the  PQ's.  There  are  certain  cases 
where  we  can  say  at  once  that  the  difference  of  resultants  is 
small,  small  that  is  relatively  to  the  whole.  This  happens 
when  all  the  P;s  are  rather  close  together,  i  e.  when  the 
component  vibrations  have  initially  nearly  the  same  phase. 
It  is  then  certain  that  at  the  end  of  the  time  t  the  amplitude 
and  phase  are  but  little  altered  from  what  they  were  at  the 
beginning.  Over  this  range  the  vibration  is  approximately 
simple,  and  the  range  is  inversely  as  the  greatest  departure 
from  the  mean  frequency  n. 
But  in  general  the  distribution  of  initial  phases  e  causes 
the  resultant  to  be  much  less  than  if  the  phases  were  in 
agreement,  and  it  may  even  happen  that  the  initial  resultant 
is  zero.  At  the  end  of  the  time  t  the  resultant  will  probably 
not  be  zero,  so  that  in  this  case  the  change  is  relatively  large. 
The  proposition  that  small  changes  in  the  phases  of  the 
components  can  lead  only  to  relatively  small  changes  in  the 
resultant  is  thus  not  universally  true  ;  and  we  must  inquire 
further  as  to  the  conditions  under  which  the  conclusion  is 
probable. 
The  most  important  case  for  our  purpose  is  when  the  initial 
phases  are  distributed  at  random,  as  they  would  presumably  be 
when  Rontgen  radiation  is  concerned.  If  the  components 
are  very  numerous  (and  of  equal  amplitude  unity),  the 
problem  is  one  which  I  have  considered  on  former  occasions*. 
It  appears  that  the  probability  of  a  resultant  amplitude  lying 
between  r  and  r  +  dr  is 
—  o-**™* 
*• C2), 
where  m  is  the  number  of  components.  Or  the  probability 
of  an  amplitude  exceeding  r  is  e~r'2'm.  The  mean  intensity 
(when  the  phases  are  redistributed  at  random  a  great  many 
times)  is  m,  corresponding  to  the  amplitude  ^/m. 
When  r  is  great  compared  with  \/m,  the  probability  of  an 
amplitude  exceeding  r  becomes  vanishingly  small.  When 
on  the  other  hand  r  is  small,  the  probability  of  a  resultant 
less  than  r  is  approximately  r*/m.  It  appears  that  the  chance 
of  the  resultant  lying  outside  the  range  from  say  ^  'm  to 
2^/m  is  comparatively  small. 
*  Phil.  Mag.  vol.  x.  p.  73  (1880);  ;  Scientific  Papers,'  vol.  i.  p.  491  ; 
'  Theory  of  Sound,1  2nd  ed.  vol.  i.  §  42  a. 
