94  M.  E.  Edlund's  Researches  on  the  Electromotive 
But  t,  or  the  excess  of  temperature  of  the  sides  of  the  copper 
cylinders,  is  unknown.  The  temperature  of  the  air  in  the  cylin- 
ders has,  of  course,  its  maximum  in  the  vicinity  of  the  wires,  and 
its  minimum  near  the  sides  of  the  cylinder.  The  temperature 
once  in  tsquilibrio  (that  is,  as  soon  as  the  cylinders  lose  an  equal 
quantity  of  heat  to  that  generated  by  the  current  through  the 
wires),  t  becomes  a  function  determined  from  the  mean  excess 
of  temperature  of  the  air  in  the  cylinders.  This  mean  excess 
results  from  the  two  following  causes  : — 1,  the  heat  generated  in 
the  wires  by  the  resistance  to  the  passage  of  the  current ;  2,  the 
variation  of  temperature  at  the  surfaces  of  contact.  We  will  call 
T  the  mean  temperature  which  would  be  produced  after  the  tem- 
perature has  arrived  at  equilibrium,  if  the  first  of  these  causes 
acted  alone;  and  we  will  designate  by  t  the  alteration  occa- 
sioned by  the  second.  For  the  case  in  which  the  two  sources 
of  heat  reinforce  one  another,  the  mean  excess  of  temperature 
above  mentioned  will  therefore  be  expressed  by  T  +  t ;  and  for 
the  case  in  which  they  are  opposed,  by  T— t. 
The  excess  of  temperature,  r,  of  the  copper  sides  is  therefore 
determined  in  the  first  case  from  T-\-t,  and  from  T— t  in  the 
second.  Now,  if  we  suppose  this  function  to  be  developed  in 
a  series  according  to  the  ascending  powers  of  T  + 1,  and  that 
sufficient  accuracy  will  be  attained  by  keeping  the  first  two 
perature  of  the  copper  cylinders,  and  M,  N,  and  a  constants.  Taking 
1  degree  Celsius  (Centigrade)  as  unit,  we  have,  according  to  Dulong  and 
Petit,  a=  1*0077  J  with  a  smaller  unit,  a  of  course  becomes  less  also.  De- 
veloping in  series,  and  designating  by  h  the  natural  logarithm  of  a,  we 
obtain 
+  *!(i+Ae  +  ^2  +  |^  +  Szc.y 
or,  abbreviating, 
^-l)-Hr+™£ +  «£  +  .*. 
Now  Jc  is  very  small;  for  a  =  1*0077  its  value  is  0-00767.  Taking  0°001 
as  unit,  we  have,  as  is  easily  demonstrated,  &= 0*00000767,  or  one  thou- 
sandth of  its  primitive  value.  This  series  is  consequently  rapidly  conver- 
gent for  values  of  r  not  too  great ;  thanks  to  which  circumstance,  keeping 
two  terms  gives  a  sufficient  approximation.  As,  further,  the  variatious  of 
0  are  relatively  small,  we  can,  without  too  great  an  error,  regard  B  as  con- 
stant. In  this  way  we  obtain  A  =  atT-\-  btT2,  a  formula  which,  by  means  of 
convenient  values  for  the  constants,  may  be  regarded  as  including  the 
term  Nt1*233. 
