148         On  the  Principle  of  the  Conservation  of  Energy. 
For  the  above  assertion  may  evidently  be  inverted,  and  we 
may  say  a  law  is  in  contradiction  with,  the  law  of  the  conser- 
vation of  force,  if  two  particles,  moving  in  accordance  with  it 
and  beginning  with  infinite  velocity,  attain,  at  a  finite  distance 
from  each  other,  finite  vis  viva,  and  thus  suffer  an  infinitely 
great  diminution  of  the  work  which  they  are  able  to  perform. 
The  two  particles  must  therefore  always  retain  an  infinite 
velocity ;  for  if  they  have  not  lost  it  in  any  finite  distance,  how- 
ever great,  they  would,  in  accordance  with  the  nature  of  potential, 
never  lose  it  even  at  greater  distances.  But  bodies  which 
always  move  relatively  to  each  other  with  an  infinite  velocity  are 
excluded  from  the  region  of  our  inquiries. 
But  if  two  particles  never  possess  more  than  finite  vis  viva} 
there  must  be  a  finite  limiting  value  of  vis  viva  which  they 
never  exceed.     It  is  consequently  possible  that  this  limiting 
ee 
value  for  two  electrical  particles  e  and  e'  may  be  =— ;  that  is, 
that  the  square  of  the  velocity,  with  which  the  two  particles 
move  relatively  to  each  other,  may  not  exceed  cc. 
The  contradiction  urged  by  Helmholtz  would,  according  to 
this,  lie  not  in  the  law,  but  in  his  assumption,  according  to 
which  the  two  particles  began  to  move  with  a  velocity  the  square 
dr* 
of  which,  namely  -p,  was  >  cc. 
If  such  a  determination  of  the  limiting  value  of  vis  viva  is 
assumed  in  connexion  with  the  law  of  the  conservation  of  force 
according  to  Helmholtz,  it  may  equally  well  be  assumed  in 
connexion  with  the  fundamental  law  of  electrical  action  (see 
section  4);  that  is,  the  work  denoted  there  by  U,  as  well  as  the 
vis  viva  denoted  by  x  (in  the  law  U  +  #  =  —  V  may  both  be  re- 
garded as  being  by  their  nature  positive  quantities. 
In  the  second  place,  it  may  be  remarked  that,  though  the 
two  electrical  particles   do  attain  infinite  vis  viva  at  a  finite 
2ee'  /l     1\ 
distance  from  each  other,  this  finite  distance  is  p  = (  -  +  -,), 
r       cc  \e     e'J 
which,  according  to  our  measures,  is  an  undefinable  small  distance, 
for  the  same  reasons  that  the  electrical  masses  e  and  e'  are  them- 
selves undefinable  according  to  our  measures.  This  distance 
was  consequently  denominated  in  section  9  a  molecular  distance. 
The  theory  of  molecular  motions  requires  in  any  case  a  special 
development,  which  as  yet  is  wanting  throughout.  But  as  long 
as  such  a  theory  remains  excluded  from  mechanical  investigations, 
any  doubts  as  to  physical  admissibility  in  relation  to  molecular 
motions  are  without  foundation. 
