302         Prof.  H.  A.  Bumstead  on  the  Heating  Effects 
cause.  In  fact  we  may  from  these  curves  get  an  approximate 
idea  of  what  the  steady  deflexions  would  have  been  if  it  had 
been  safe  to  continue  running  the  bulb  at  the  rate  necessary 
for  such  large  deflexions. 
If  \  is  the  coefficient  of  absorption  of  the  metal  for  the 
rays  used,  then  the  energy  of  the  rays  at  any  point  in  the 
interior  of  the  strip  whose  distance  is  x  from  the  front  face 
is  I0e~Xx  and  the  energy  absorbed  in  an  element  of  thickness 
dx  is  \l0e-Xxdx.  Let' us  assume  that  the  heat  generated  in 
the  element  is  proportional  to  this,  say  aXl0e-Xxdx.  The 
equation  of  the  flow  of  heat  under  these  circumstances  is  : 
cdt=*d^+aXloe 
with  the  boundary  conditions 
,(^)      =AVo;     -*(§)      =W,; 
\ax  Jx=o  \dd''x=i 
where  V  is  the  temperature  (measured  above  the  sur- 
roundings), t  the  time,  c  the  specific  heat  of  unit  volume,  k 
the  conductivity,  h  the  emissivity  of  the  surface,  and  /  the 
thickness  of  the  strip.  The  solution  of  this  will  have  the 
form 
where  Vw  is  the  steady  value  and  is  the  solution  of  the 
differential  equation  with  the  left-hand  member  put  equal  to 
zero;  the  successive  values  of  7  in  the  sum  are  determined 
by  a  certain  transcendental  equation,  and  the  A's  are  functions 
of  x  into  which  7  enters.  What  is  observed  is  the  temperature 
of  the  surface  where  x  —  l.  The  variation  of  this  temperature 
with  the  time  is  represented  fairly  well  (except  at  the  very 
beginning)  by  a  single  term  of  the  sum  of  exponentials,  so 
that  we  may  write  as  a  rough  approximation 
The  variation  of  the  temperature  with  the  time  during 
cooling  will  have  different  values  of  the  A's  and  7's,  but 
(again  excepting  the  initial  part)  the  curves  of  heating  and 
cooling  will  have  very  nearly  the  same  form.  This  is  seen 
in  the  figure,  and  it  also  appears  mathematically  that,  with 
the  values  of  the  conductivity  and  emissivity  involved,  there 
is  a  nearly  uniform  temperature  throughout  the  metal  within 
less  than  a  minute  after  the  rays  are  turned  on.  We  may 
therefore,  for  such  accuracy  as  is  needed  in  this  discussion. 
