On  Resonance,  and  Change  of  Phase  accompanying  Reflection,    201 
been  given,  it  appears  that  Laplace's  and  Gauss's  second  demon- 
stration depend  on  principles  the  general  truth  of  which  is  ap- 
parent on  consideration,  and  the  mathematical  difficulties  are  not 
of  a  very  serious  kind.  The  illustration  derived  from  the  deflec- 
tion of  a  stone  let  fall  on  to  a  plane,  first  given  by  Sir  John 
Herschel  in  the  Edinburgh  Review,  has  been  sometimes  regarded 
as  a  proof  of  the  law*.  So  far  from  beiug  self-evident,  it  appears 
to  me  that  the  assumption  there  made  of  independent  x  and  y 
deflections  is  one  of  the  most  striking  consequences  of  Laplace's 
law  of  facility.  A  priori,  one  would  be  inclined  to  think  that 
the  largeness  of  an  x  deflection  would  increase  the  probability 
of  a  small  y  deflection,  as  then  the  total  deflection  would  be  ren- 
dered less.  At  all  events,  the  independence  of  the  x  and  y  de- 
flections is  a  very  unexpected  result  when  it  is  considered  that 
it  may  be  derived  as  a  consequence  of  the  sole  assumption  that 
errors  that  occur  are  due  to  the  aggregation  of  errors  arising 
from  a  great  number  of  sources.  "With  reference  to  HerschePs 
problem,  Ellis  has  remarked  that  "  there  is  no  shadow  of  reason 
for  supposing  that  the  occurrence  of  a  deviation  in  one  direction 
is  independent  of  that  in  another,  whether  the  two  directions  are 
at  right  augles  or  not"  (Phil.  Mag.  vol.  xxxvii.  (1850)  p.  3.25). 
XXIII.  On  Resonance,  and  on  the  Circumstances  under  which 
Change  of  Phase  accompanies  Reflection.  By  Robert  Moon, 
M.A.,  Honorary  Fellow  of  Queen's  College,  Cambridge-^. 
TN  a  former  paper  {  I  pointed  out  the  misapprehensions  which 
have  arisen  with  regard  to  the  resonance  which  occurs  when 
a  disk  or  tuning-fork  is  made  to  vibrate  near  the  open  end  of  a 
tube  of  which  the  other  extremity  is  closed,  and  whose  length  is 
equal  to  one  quarter  of  the  length  of  a  wave  having  the  same 
periodic  time  as  the  disk  or  fork;  and  I  showed  that  the  aug- 
mentation of  sound  which  under  those  circumstances  results  is 
due  to  the  fact  that  the  aerial  disturbance  caused  by  each  move- 
ment in  either  direction  of  the  vibrating  body,  by  reflection  at 
the  closed  end  of  the  tube,  is  brought  to  bear  on  the  disk  or  fork 
during  its  movement  in  the  opposite  direction,  so  as  to  increase 
the  amplitude  of  its  vibration. 
If  the  disk  and  tuning-fork  are  replaced  by  a  bell  which  can 
be  made  to  vibrate  by  means  of  a  bow  drawn  across  its  rim,  and 
the  width  of  the  tube  is  at  the  same  time  enlarged,  its  length 
continuing  equal  to  one  quarter  of  the  length  of  a  wave  of  the 
*  As  in  Thomson  and  Tait's  '  Natural  Philosophy,'  vol.  i.  p.  314. 
t  Communicated  by  the  Author. 
X  "  On  a  Simple  Case  of  Resonance,"  inserted  in  the  Philosophical  Ma- 
gazine for  February. 
