134  Prof.  W.  Weber  on  Electricity  in  relation  to 
The  substitution  of  these  values  gives 
2  2£  +2r ^  =  2(w2  +  w'2~2m/  cos  e)  -f  2r(N  cos  v-  N'  cos  v'), 
dr2 
3  -^  =S(u  cos  ©— w'  cos  ©)2. 
The  second  equation  subtracted  from  the  first  gives 
dr  ddr 
-  ^  +2r^  =2{u*  +  u!*-2uu'  cos  e) -3(« cos  6-^ cos  ©')2 
+  2r(Ncosj/— N'cosv'), 
whence  the  general  expression  for  the  repelling  force  of  two  elec- 
trical particles  e  and  e'  at  the  distance  r,  namely 
ee/.       1  dr2  .  2d  dr 
rr 
/J.  d^2ddr\ 
\        cc  dt*+ cc  dt*)' 
is  obtained  in  the  following  transformed  shape, 
ee' 
= |>  +  2(w2  +  w'2— 2ww'cose)— 3(wcos©  —  w'cos©)2 
CC// 
+2r(N  cos  v~N'  cos  v')] . 
By  substituting  for  the  particle  e'  the  positive  electricity  in 
the  given  element  of  current,  namely  +e'ds,  this  expression 
gives  the  repelling  force 
ee  ds 
—[cc  +  2(u*  +  u!2—2uu'  cos  e)  —  3(w  cos  ©  — v!  cos  ©')2 
+  2r(N  cos  v—  N'  cos  v') ] ; 
but  by  putting  for  the  particle  e'  the  negative  electricity  in  the 
given  element  of  current,  namely,  —e'ds1,  we  obtain  the  repelling 
force 
ee  ds 
[—cc—2(u*  +  u'2  +  2uu'  cos  e)  +  3(w  cos  ©  +  w'  cos  ©)2 
ccrr  L 
— 2r(N  cos  v  +  N'cos j/)], 
since  in  this  case  e  +  7r,  ©'  +  7r,  and  v'  +  tt  take  the  place  of  e, 
©',  and  j/;  and  these  therefore  give  together  the  total  repelliug 
force  between  the  moving  particle  e  and  the  whole  element  of 
current,  namely 
4>pp'd\ 
(3W  cos  ©  cos  <&,—2uu'  cos  €— rW  cos  v) . 
ccrr 
The  repelling  force  between  the  stationary  particle  —  e  and  the 
whole  element  of  current,  on  the  other  hand,  if  r  denotes  the 
distance  of  the  stationary  particle  —  e  from  the  given  element  of 
