138  Prof.  W.  Weber  on  Electricity  in  relation  to 
We  also  get  from  (2)  and  (3), 
-2arKd{cosv)=d(u2r2) (7) 
The  integration  of  the  differential  equation  (6)  gives,  after  mul- 
tiplying by  2  and  putting  V=  —  I 1  J, 
2arcosv=  —  I lj  — aa— uu  +  const. ;    .     .      (8) 
and  from  this,  since  r=r0,  oi  =  ct0}  and  cosv=— 1  when  w  =  0, 
comes 
pec  ,„* 
— 2«r0=  — a0a0+  const.       ...      (9) 
ro 
Equation  (9),  subtracted  from  equation  (8),  gives 
2ar  cos  v -+■  2ar0=  ( -  —l\uu  +  pccl— )-aa-fa0a0.  (10) 
By  integrating  the  differential  equation  (7)  we  obtain,  after  divi- 
ding by  r3, 
otx         Cctctdr 
—2acosv= \-6  1 -j 
r         J    rr 
or,  multiplying  by  r, 
—  2arcosv  =  aa  +  3r  I ,     .     .     .      (11) 
J    rr 
and  hence,  for  the  sum  of  (10)  and  (11), 
'uadr 
rr 
and  therefore 
2flr0=^  -lJMM  +  pcc^- -  -)  +  «0*o+  3rJ  - 
«tt=  — -  (pw(~  -l)+r«0a0+  3rr  (^  -2«r0rj.  (12) 
From  equation  (3)  there  follows  further,  since  dr=udt, 
dv=  ■ 
and  since,  by  equation  (7), 
*=-*; (13) 
u  r  v     ' 
(COS  V)  = s — *    > 
2«r* 
and  by  equation  (11), 
1  /aa  •      f  a2^r\ 
cos"=-2-«(-7+3J^> 
we  get,  by  substituting  these  values  in  the  identical  equation 
7  d  (cos  v) 
av= 
*/(]  —  cosv2)' 
