see Figure "∂[∂(t): Aθpc(POIo(t),t)]" 

For small angles d[theta] :   
(1)	chord  = radius*d[theta]   
	= Rpcs(POIo(t),t)*d[Aθpc(POIo(t),t)]
But also from the Figure :
	chord 
	= Vons(PART)*d[t]*cos(Aθpc(POIo(t),t) + PI/2)  ...where d[theta] is a differential change in O
But : 	cos(alpha) = sin(alpha + PI/2) 
so : 
(2)	chord 
	= Vons(PART)*d[t]*sin(Aθpc(POIo(t),t))  ...where d[theta] is a differential change in O

Equating (5) & (6) :
	Rpcs(POIo(t),t)*d[Aθpc(POIo(t),t)] =  Vons(PART)*d[t]*sin(Aθpc(POIo(t),t))
or : 

(mathH)  ∂[∂(t): Aθpc(POIo(t),t)] =  Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)
(endMath)



/*+-----+
Reminders for Chapter 4 : 
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)  
and 
	(5)	sin(Aθpc(POIo(t),t)) 
	=    Rocs(POIo)*sin(Aθoc(POIo)) 
	  /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)  

Subbing (2)* & (5)* into (3)  : 
	∂[∂(t): Aθpc(POIo(t),t)] 
	=  Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)
	=  Vons(PART)
	  *Rocs(POIo)*sin(Aθoc(POIo))
	  /{ 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)  

Finally : 
(mathH)  ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) / {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 (radians/time)

/%As Aθpc(POIo(t),t) -> 0 :  
L1)	∂[∂(t): Aθpc(POIo(t),t)]  
	->     Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) 
	  /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)
	->     Vons(PART)*Rocs(POIo)*0
	  /{ Rocs(POIo)^2 - 2*Rocs(POIo)*1              *Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)
	->  0 
/*This is OK, as when the particle is far beyond the coordinate center (t is very large), eventually the rate of anular changes go to zero.

/%As Aθpc(POIo(t),t) -> PI :  
L2)	∂[∂(t): Aθpc(POIo(t),t)]  
	->     Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) 
	  /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)
	->     Vons(PART)*Rocs(POIo)*0
	  /{ Rocs(POIo)^2 - 2*Rocs(POIo)*(-1)           *Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)
	->  0 
/*This is OK, as when the particle is far behind of the coordinate center (t is large negative), eventually the rate of anular changes go to zero.

/%As Aθpc(POIo(t),t) -> PI/2 :  
L3a)	∂[∂(t): Aθpc(POIo(t),t)]  
	->     Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) 
	  /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)
	->     Vons(PART)*Rocs(POIo)*1
	  /{ Rocs(POIo)^2 - 2*Rocs(POIo)*0              *Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)
	->     Vons(PART)*Rocs(POIo)
	  /{ Rocs(POIo)^2                                                 + [Vons(PART)*t]^2 }^(1/2)
Check : 
L3b)	see Figure "Calculus for a POIo"
	∂[∂(t): Aθpc(POIo(t),t)] 
	=  Vons(PART) / Rpcs(POIo(t),t) 
	=  Vons(PART) 
	  / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)   
	=  Vons(PART) 
	  / { Rocs(POIo)^2 - 2*Rocs(POIo)*0              *Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)   
	=  Vons(PART) 
	  / { Rocs(POIo)^2                                                 + [Vons(PART)*t]^2 }^(1/2)   

/*WRONG!! -  missing Rocs(POIo) in the numerator