Force  in  the  Contact  of  Metals.  95 
terms  (a  supposition  which  ought  to  be  verified  by  observation), 
we  obtain:— in  the  first  case,  r=y(T  +  t)  +  S(T+/)2;  in  the 
second,  T  =  y(T  —  t)-\-S[T  —  ty2, — where  7  and  8  are  constants. 
Substituting  these  values  iu  the  above  expression  gives : — 
Ay=«(T  +  0+Z>(T  +  02,     Aw=«(T--0  +  5(T-*)2,    .     (1) 
Al-All  =  2at  +  4bTt, (2) 
where  a  and  b  are  new  constants. 
But  Ay  — Aw  is  only  the  difference  between  the  quantities  of 
heat  produced  when  the  current  passes  first  in  one  direction  and 
then  in  the  other — or,  in  other  terms,  twice  the  quantity  of  heat 
(W)  produced  or  destroyed  at  the  point  of  contact,  and  which  is 
precisely  the  quantity  sought  by  these  observations.  We  have 
therefore 
Vr=at  +  2bTt .     (3) 
T  was  not  measured;  its  value  is  consequently  unknown;  t 
was  obtained  directly,  from  the  movement  of  the  index  when 
the  current  passed  first  in  one  direction  and  then  in  the  other. 
As  we  have  seen  above,  T  represents  the  excess  of  temperature 
which  would  be  produced  in  the  air  enclosed  in  the  copper  cy- 
linders if  the  current  traversed  the  pair  of  wires  soldered  together 
without  any  variation  of  temperature  taking  place  at  the  point 
of  soldering.  If,  now,  m  designates  a  constant  proportional  to 
the  resistance  of  the  pair  of  soldered  wires,  the  heat  developed 
in  the  same  pair  of  wires  by  the  current,  s,  is  equal  to  ms2.  We 
have  then,  according  to  equation  (1) : — 
ms2  =  aT  +  bT* (4) 
If  now  in  equation  (3)  the  value  of  T  derived  from  equation 
(4)  be  substituted,  remembering  at  the  same  time  that  W  is 
proportional  to  the  intensity  of  the  current,  and  consequently 
can  be  expressed  by  ns,  where  n  is  a  constant  factor,  we  obtain 
™_       /ms1*       a* 
/9,  we  obtain,  finally, 
us={V$s*+l)t (5) 
In  this  equation,  a  is  proportional  to  the  quantity  of  heat 
produced  or  destroyed  at  the  point  of  contact  at  the  time  of  the 
passage  of  a  current  of  unit  intensity,  t  designates  the  displace- 
ment of  the  index  when  the  current  passes  from  one  direction  to 
the  other,  and  /3  is  a  constant  which  changes  its  value  when  one 
Afb?7i 
By  reducing  this  equation,  and  designating  -  by  «,  and  — =-  by 
a' 
