14  Prof.  W.  Weber  on  Electricity  in  relation  to 
6.  Extension  of  the  Principle  of  the  Conservation  of  Energy  to 
two  electrical  particles  which  do  not  form  a  detached  system. 
If  potential  energy  is  taken,  as  is  done  in  the  previous  section, 
as  equal  and  opposite  to  potential,  the  principle  of  the  conserva- 
tion of  energy  holds  good  for  two  particles  only  so  long  as  these 
two  particles  constitute  a  detached  system — that  is,  so  long  as  the 
system  formed  of  the  two  particles  undergoes  neither  gain  nor 
loss  of  energy  from  without. 
If  the  total  energy  of  such  a  detached  system  of  two  particles 
were  at  first  =  A,  but,  the  system  ceasing  to  be  detached,  it  re- 
ceived from  without  a  quantity  of  kinetic  energy  =  a,  it  seems 
to  follow  that,  if  the  system  were  now  again  to  become  detached, 
the  total  energy  would  again  become  and  remain  constant  so 
long  as  it  remained  detached,  but  that  the  total  energy  of  the 
system  in  its  final  detached  state  would  have  the  value  A  +  a 
(that  is,  a  value  exceeding  that  corresponding  to  its  previous  de- 
tached state  by  a).  This,  however,  does  not  by  any  means  con- 
clusively prove  the  impossibility  of  extending  the  principle  of 
the  conservation  of  energy  to  two  electrical  particles  which  do 
not  constitute  a  detached  system. 
For,  strictly  speaking,  this  has  only  been  proved  on  the  as- 
sumption that  the  potential  energy  of  the  system  depends  solely 
on  the  distance  between  the  two  particles ;  while  if,  on  the 
other  hand,  the  potential  energy  does  not  depend  simply  on  the 
distance  of  the  two  particles,  but  also  on  their  relative  motion, 
it  is  evident  that  while  the  system  receives  from  without  an 
amount  of  kinetic  energy  =  a,  a  change  in  its  potential  energy 
may  be  indirectly  produced  thereby.  It  is  thus  possible  that 
the  change  of  potential  energy,  so  caused  indirectly  from  without, 
might  be  =—a,  so  that  the  total  energy  (kinetic  energy  and 
potential  energy  together)  of  the  two  particles,  even  if  they  did 
not  constitute  a  detached  system,  would  retain  always  the  same 
value. 
This,  however,  certainly  does  not  occur  in  reality  for  a  system 
.  of  two  electrical  particles,  if  the  potential  energy  is  taken  as  equal 
and  opposite  to  the  potential ;  but  this  assumption,  which  would 
thus  make  the  extension  of  the  principle  impossible,  has  by  no 
means  been  proved  to  be  a  necessary  one.  In  general,  all  that 
is  required  is  a  special  determination  of  the  way  in  which  the  po- 
tential energy  depends  upon  the  potential;  and  here,  all  that  is 
self-evident  is,  that  inasmuch  as  potential  and  potential  energy 
are  homogeneous  magnitudes,  a  purely  numerical  relation  must 
exist  between  them.  But  whether  this  numerical  relation  is 
always  that  of  +1  to  —  1,  or  whether  it  is  to  be  fixed  other- 
wise, must  still  be  regarded  as  in  general  doubtful ;  so  that  the 
possibility  of  the  extension  of  the  principle  still  remains. 
