the  Principle  of  the  Conservation  of  Energy.  137 
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of  the  acceleration  —  which  depends  upon  the  segregating  force , 
from  the  total  value  — }  the  difference  l-j-  —a  sin  v  J  gives  that 
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part  of  the  total  acceleration  —  which  results  from  the  existing 
Oil 
motion  under  the  sole  influence  of  the  forces  exerted  upon  each  other 
by  the  two  particles.  But,  under  the  sole  influence  of  the  attract- 
ing or  repelling  forces  exerted  upon  each  other  by  the  two  par- 
ticles, the  element  of  surface  ocrdt,  described  in  a  given  element 
of  time  dt,  would  have  a  constant  value,  or  we  should   have 
«-r  +r  —  =  0;  hence  the  resulting  part  of  the    acceleration 
-i-  becomes 
_«  dr_t 
r  dt' 
By  equating  this  part  with  the   above  difference,  we  get  the 
equation 
dot  .  oc  dr 
-j  —asinv  = — • 
dt  r  dt 
Besides  these,  we  have,  as  is  self-evident,  a  third  equation, 
7        ocdt 
dv=  — • 
r 
Accordingly,  for  the  four  variable  magnitudes  r,  u,  a,  v,  there 
are  the  following  three  equations  : — 
oca.      pec    dV      du  .... 
acosv-T  =  2^>-lfr-di>   •■••<!) 
udr      du 
asmv-7dt=dt> W 
*-? (3) 
Multiplying   equation    (1)   by   dr  =udt,   and  equation    (2)    by 
rdv  =  adt,  we  obtain 
7        aadr      pec      dV  7  . 
acosv.dr--—  =  ^-j'-j-dr^udu,     .     .       (4) 
r 
2ee'     dr 
ar&inv.dv =  ada (5) 
r 
The  difference  of  these  two  equations  gives 
a  .d(r  cos  v)=^— p  -j-  dr—udoc—udu.        .     .     (6) 
