the  Principle  of  the  Conservation  of  Energy.  133 
particle  —  e'ds.  Also,  let  r  denote  the  distance  of  the  element 
of  current  from  the  particle  e3  u  the  velocity  of  the  particle  e, 
x,  y,  z  the  coordinates  of  the  particle  e,  x\  y1,  z'  the  coordinates 
of  the  element  of  current,  0  and  0'  the  angles  which  the  di- 
rections of  u  and  u1  make  with  r,  and  e  the  angle  between  the 
directions  of  u  and  u1. 
Next,  let  the  general  expression  for  the  repelling  force  of  two 
electrical  particles  e  and  e1  at  the  distance  r,  namely 
rr  \        cc 
<h*      2r  ddr\ 
dt*  +  cc  dt*)9 
be  transformed  as  follows  (see  Beer,  Einleitung  in  die  Elektro- 
statik,  die  Lehre  vom  Magnetismus  und  die  Electrodynamik, 
S.  251).     First,  let  the  equation 
rr=(x-x{)*+(y-y')*  +  {z-z!)* 
be  differentiated  with  respect  to  the  time  t ;  we  then  get 
or  also 
dr 
r  -j-  =  r (u  cos  0  —  v!  cos  ©') . 
By  a  second  differentiation  we  get 
dr*        ddr_ 
dt*  +r  dt* 
ddr^_/dx_drf_S*     (ty_djh\     ,  (dA_(M}\ 
\dt        dt)   +  \dt       dt)   *\dt       dt) 
wherein 
(dx      dod\*      idy      dy\*  ,  (dz      dz\*       2  ,    ,2     0     . 
If  now  the  acceleration  of  the  one  particle,  whose  components 
are  -j-%,  --^-,  -—%-,  be  denoted  by  N,  and  the  angle  which  its  di- 
rection makes  with  r  by  v,  and  in  like  manner  the  acceleration 
.  ddx'    ddy1   ddz1 
of  the  other  particle,  whose  components  are     ,g  ,  —r|->  ~ja>  dv 
N'j  and  the  angle  which  its  direction  makes  with  r  by  v\  we 
obtain 
x  —  zJ/ddx       ddx*\      y—y'/ddy      ddy'\      z—z'  /ddz  __ddz'\ 
=  N  cos  v  — N'  cos  i/. 
