Let  O  X  and  O  Y  be  two  rectangular  co-ordinate  axes  to  which  all
points  of  the  cross-section  are  referred.
Represent  by

L,  the  original  length  of  an  elementary  fibre  as  D  B,  a  b,  (Fig.  A)

a  =  dx  dy  the  area  of  its  cross-section  ;

x  and  y,  the  co-ordinates  of  a ;

a,  the  infinitely  small  angle  which  the  plane  C'  D'  makes  with
its  original  position  C  13  after  deflection.
Now,  from  the  hypothesis  adopted,  any  fibre,  as  a  b  (Fig.  A),
contained  between  two  consecutive  planes,  will,  after  deflection,  be
lengthened  by  an  amount  equal  to  b  O  in  the  relative  change  of
position  of  the  plane  C  D  ;  and  as  the  distance  of  this  fibre  from
the  neutral  axis  is  y,  this  increase  of  length  will  be  expressed  by,


in  like  manner,  the  decrease  in  length  of  any  fibre  at  the  same  dis
bance  from  the  neutral  axis,  on  the  other  side  of  it,  will  also  be  ex
pressed  by  y  «.
JO  2  APPENDIX.
Resuming  now  Eq.  (A),  and  substituting  in  its  second  membra
dx  dy  =  a  for  A,  and  y  a.  for  Z,  there  obtains

E  dx  dy  4^,
LJ

which  expresses  the  relation  between  the  strain,  and  the  correspond-
ing elongation  for  any  elementary  fibre.
Therefore   the  total  strain  on  the  fibres  elongated  will  be  ex-
pressed by


In  like  manner  the  strains  on  the  compressed  fibres  will  be  ex-

_JL


the  negative  sign  being  used  to  denote  the  contrary  direction  of
the  elastic  resistance  of  the  compressed  fibres.
