∂[∂(t):  Rpcs(POIo(t),t)] 
		= ∂[∂(t): |Rpcv(POIo(t),t)|]
		=  Rpcv(POIo(t),t)     dotPRod    ∂[∂(t): Rpcv(POIo(t),t)] / |Rpcv(POIo(t),t)|
from (1) above in "∂[∂(t): Rpcv(POIo(t),t)]" 
	∂[∂(t): Rpcv(POIo(t),t)] =  -Vonv(PART)
therefore :
 	∂[∂(t): Rpcs(POIo(t),t)] 
	=  Rpcv(POIo(t),t)     dotPRod           -Vonv(PART) / |Rpcv(POIo(t),t)|
	= |Rpcv(POIo(t),t)| *cos(Aθpc(POIo(t),t))*|-Vonv(PART)|/ |Rpcv(POIo(t),t)|
	= -Vons(PART)		  *cos(Aθpc(POIo(t),t))

(1)	∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]  = -Vons(PART)*cos(Aθpc(POIo(t),t))


/*CONFIRMATION :  
Show Figure "∂[∂(t): Rpcs(POIo(t),t)]" 

/%(2)	|Rpcv(POIo(t),t)| = Rpcs(POIo(t),t)
	=  {   [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2 }^(1/2)

(3)	   ∂[∂(t): Rpcs(POIo(t),t)] 
	=   1/2*{  [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) ]^2 }^(-1/2) 
	  *∂[∂(t): {(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2}]

(4)	   ∂[∂(t): {(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2}]
	=  ∂[∂(t): [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2]
	 + ∂[∂(t): {Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))}^2]
/*But for all t,  the following are constants, as Vod(POI) is parallel to the 
coordinate axis [rO0och, rO0pch] : 
/%	  Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))
	= Rocs(POIo)*sin(Aθoc(POIo))
	= constant over t for any (POIo), (POIp)
therefore 
	   ∂[∂(t): {(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2}]
	=  ∂[∂(t): [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2]
	=  2                                              *[Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) ]
	                                           *∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]

/*Where from "∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) ]" below :
/%	∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] =  -Vons(PART)

therefore 
(5)	   ∂[∂(t): {(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2}]
	= -2*Vons(PART)                               * Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))

Repeating (3)
	∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIp)|] 
	= 1/2*{    [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) ]^2 }^(-1/2) 
	  *∂[∂(t): [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))]^2]
Substituting (5) into (3) : 
	= 1/2*{    [(Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]^2 + [Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) ]^2 }^(-1/2) 
	 -2*Vons(PART)                                * Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))  
	=  -Vons(PART)                                * Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) 
	         /Rpcs(POIo(t),t) *{ sin(Aθpc(POIo(t),t))]^2 +                cos(Aθpc(POIo(t),t)) ]^2 }^(-1/2) 

But 1 = { sin(A0oc(POIp(t),t))^2 + cos(A0oc(POIp(t),t))^2 }^(-1/2)
therefore : 

(6)	∂[∂(t): Rpcs(POIo(t),t)] =  -Vons(PART)*cos(Aθpc(POIo(t),t))

/*This is the same as (1) and (expr) above, confirming that result.


Actually, there is NO essential difference other than sign between the derivations for :
/%
	∂[∂(t): Rpcs(POIo(t),t)] = ∂[∂(t): |Rpcv(POIo(t),t)|]  =  -Vons(PART)*cos(Aθpc(POIo(t),t))
	∂[∂(t): Rocs(POIo)]		 = ∂[∂(t): |Rocv(POIp(t),t)|]  =   Vons(PART)*cos(Aθoc(POIp(t),t))
/*which surprises me a bit, but it shouldn't be a surprise!!


/*********************
OBJECTIVES :  Where convenient, express variables as functions of FIXED variables :
(POIp) ->  -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp)
(POIo) ->   Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo)


/*From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,Aθpc,Aθpc]@t  to [Rocs,Rθ0ocs,RθPI2ocs,Aθoc,APo] for (POIo)"
/%
	(2)	Rpcs(POIo(t),t))
	=  {    Rocs(POIo)^2
	    - 2*Rocs(POIo)*cos(Aθoc(POIo)) *Vons(PART)*t
	    +                              [Vons(PART)*t]^2 
	   }^(1/2)  

Therefore
	   ∂[∂(t): Rpcs(POIo(t),t)] 
	=  ∂[∂(t): {Rocs(POIo)^2
	            - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t
	            +                             [Vons(PART)*t]^2 
	           }^(1/2)  
	     ]
	=  1/2*    {    Rocs(POIo)^2
	            - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t
	            +                             [Vons(PART)*t]^2 
	           }^(-1/2)
	  *∂[∂(t): Rocs(POIo)^2
	            - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t
	            +                             [Vons(PART)*t]^2 
	     ]
	=  1/2*    {    Rocs(POIo)^2
	            - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t
	            +                             [Vons(PART)*t]^2 
	           }^(-1/2)
	 *{  ∂[∂(t): Rocs(POIo)^2] 
	   + ∂[∂(t):  - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t] 
	   + ∂[∂(t): [Vons(PART)*t]^2]
	   }
	=  1/2*    {    Rocs(POIo)^2
	            - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t
	            +                             [Vons(PART)*t]^2 
	           }^(-1/2)
	 *{  0                                    ] 
	   - 2*Rocs(POIo)*cos(Aθoc(POIo)) *Vons(PART)
	   + 2*Vons(PART)*t*Vons(PART)
	   }
	=  {-   Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART) + Vons(PART)^2*t }
	  /{    Rocs(POIo)^2
	    - 2*Rocs(POIo)*cos(Aθoc(POIo))                *Vons(PART)*t
	    +                                             [Vons(PART)*t]^2 
	   }^(-1/2)
	=		Vons(PART)*{- Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t}
			/{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(-1/2)


then :                     
(mathH)
	∂[∂(t): Rpcs(POIo(t),t)]
	=		Vons(PART)*{- Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t}
			/{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(-1/2)
(endMath)



/*+-----+
LIMIT CHECKS :  

Dimensional consistency -  OK, as all terms reduce to (length/time).

I haven't yet shown equivalence between the "-Vons(PART)*cos(Aθpc(POIo(t),t))" result and equation (7)...

See  "∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))]" - note that the ?correct? result as obtained by the "Alternative Verification" provides a small degree of confirmation of the results for this sub-sub-section.  


As t -> +- infinity : 

As t -> +- 0 :

/%As cos(Aθoc(POIp(t),t)) -> 0  : 

/*endsection