the  Principle  of  the  Conservation  of  Energy.  14? 
Helmholtz  deduces  from  the  law  of  electrical  force  the  equation 
of  motion  of  two  electrical  particles  for  motions  in  the  direction 
of  the  distance  r  of  the  two  particles,  namely 
1     dr2  r 
cc     dt2    ,  ed 
\mcc  —  — 
r 
pj  Qpf! 
or,  putting  C  =  —  and =  p,  the  equation 
rQ         mcc 
1  dr2_r  —  r0  ^  p  m 
cc  dt2     r—p     r0 ' 
that  is  to  say,  the  same  equation  as  was  arrived  at  in  section  9. 
If—  >±mcc>C— that  is,  if  ^>1  >£,  we  have  —  positive  and 
r      -  r  r0}  dfv 
greater  than  cc,  and  -j-  is  therefore  real.     If  the  latter  is  also 
.    ed 
positive,  r  will  increase  until  —  =|mcc,  that  is  till  r=p,and  then 
j-  becomes  infinitely  great. 
.  .  ed 
The  same  will  happen  if,  to  begin  with,  C  >\mcc>  — ;  that  is, 
if  "  >1  >c    and  ._  is  negative. 
r0  r  dt  ° 
These  consequences  are,  according  to  Helmholtz,  in  contra- 
diction with  the  law  of  the  conservation  of  force. 
Now  it  may  be  remarked  hereupon,  in  the  first  place,  that 
two  electrical  particles  are  here  assumed  which  begin  to  move 
with  a  finite  velocity  certainly,  but  one  which  is  greater  than 
.ii-  t    .   *      .i        i^^A^n    ,~r  millimetre 
the  velocity  c — greater,  that  is,  than  4394o0 .  10° 1—  • 
J         °  '  second 
The  case  of  two  bodies  moving  relatively  to  each  other  with  such 
a  velocity  is  nowhere  recognizable  in  nature.     In  all  practical 
1   dr2 
cases  we  are  accustomed  rather  to  treat rs  as  a  very  small 
cc  dt2  J 
fraction ;  and  this  deserves  notice. 
For,  according  to  Helmholtz  (loc.  cit.  p.  7),  a  law  is  in  con- 
tradiction with  the  law  of  the  conservation  of  force  if  two  par- 
ticles, moving  in  accordance  with  it  and  beginning  with  a  finite 
velocity,  attain,  within  a  finite  distance  of  each  other,  infinite  vis 
viva,  and  so  are  able  to  do  an  infinitely  great  amount  of  work. 
The  principle  seems  to  be  here  announced  that,  according  to 
the  law  of  the  conservation  of  force,  two  particles  cannot,  under 
any  circumstances,  possess  infinite  vis  viva. 
L2 
