Lucas   
/$		F(r,v) = q*{ET(r,v) + (v/c)Bi(r,v)}  
		=   q* E0(r) * (1 - β^2) /(1 - β^2*sin(θ´)^2)^(3/2)				*[(1 - β^2      + β^2*cos(θ´)^2)*r     - (β•r)*r(rβ)]  
		=   q* E0(r) * (1 - β^2) /(1 - β^2*sin(θ´)^2)^(3/2)  				*[(1            - β^2*sin(θ´)^2)*r     - (β•r)*r(rβ)]  
		=   q* E0(r) * (1 - β^2) /(1 - β^2*sin(θ´)^2)^(1/2) - q*|E0(r)|* (1 - β^2) /(1 - β^2*sin(θ´)^2)^(3/2)  *(β•r)*r(rβ)  

/*	Can one say?  :  
	where the first term is conventional (Maxwellian-Relativity)  
	and the second is NEW from Lucas's Universal Force   

/*	Howell - starting with (4-26) 
	Derived Lorentz Force F_L(const v : q,E,Bi) 
/$ 		F(r´,t´) = q*ET(r - v*t,t) + q/c*[vBi(r - v*t,t)]  

/*  Lucas puts into observer frame of reference
/$1)	F(r,v) = q*ET(r,v) + q/c*[vBi(r,v)]  

/*  Using (4-43) E&B_fields_self_consistent 
/$1a)	ET(r,v)  =  E0(r)*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)  
		Bi(r,v) =  (v/c)ET(r,v)   

/*  Putting (1a) terms into (1) :
/$		F(r,v) 
		=  q*E0(r)*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2) + q/c*{ v[ v/cE0(r)*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2) ] }   
		=  q		 *(1 - β^2)/(1 - β^2*sin(θ)^2)^(3/2) *[ E0(r) + 1/c*v[ v/cE0(r) ]      
2)		=  q		 *(1 - β^2)/(1 - β^2*sin(θ)^2)^(3/2) *[ E0(r) +   v/c[ v/cE0(r) ]      

/*  Consider (2) - the expression (v/c)[(v/c)E0(r)]
   Lucas provides a corresponding vector identity in (4-45) below (this is verified later) : 
   04_45 Vector identities for Lorentz Force derivation  

/$3)	(v/c)[(v/c)E0(r,v)]  =  (v/c)•E0(r)*[(v•rh)*rh/c - rh(rhv)/c] - (vs/c)^2*E0(r)   
/*  Subbing (3) into (2) : 
/$		F(r,v) 
		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2) *{ E0(r) + (v/c)•E0(r)*[(v•rh)*rh/c    -             rh(rhv)/c] - (vs/c)^2*E0(r) }      
4)		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2) *{ E0(r) + (v/c)•E0(r)* (v•rh)*rh/c    - (v/c)•E0(r)*rh(rhv)/c  - (vs/c)^2*E0(r) }      

/* looking at term : 
/$		  (v /c)•E0 (r)        *(v    •rh         )*rh/c
		= (v /c)•E0 (r)        *(v /c •rh         )*rh
		= (vs/c)*E0s(r)*cos(θ´)*(vs/c*|rh|*cos(θ´))*rh
		= (vs/c)       *cos(θ´)*(vs/c*1    *cos(θ´))*rh*E0s(r)
		= (vs/c)^2*cos(θ´)^2*rh*E0s(r)
/* but 
/$		rh*E0s(r) = E0(r), vs/c = β 
/*	so 
5)		???? = b^2*cos(Op)^2*E0(r)

/*	looking at term :
/$		(v/c)•    E0 (r)*rh(rh v)/c
		= (v/c)•rh*E0s(r)*rh(rh(v /c))	ET(r,v)  =  E0(r)*(1 - β^2)/(1 - β^2*sin(θ)^2)^(3/2)  
			
/*	Now setting b_v = b*v_h = v/c
/$6)	= E0s(r)*(b_v•rh)*rh(rhb_v)

/* subbing (5) and (6) into (4) : 
/$4)	F(r,v) 
		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*{ E0(r) + (v/c)•E0(r)* (v•rh)*rh/c - (v/c)•E0(r)*rh(rhv)/c       - (vs/c)^2*E0(r) }      
7)		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*{ E0(r) + β^2*cos(θ´)^2*E0(r)        - E0s(r)*(b_v•rh)*rh(rhb_v)  - (vs/c)^2*E0(r) }      

/*	NOTE :  from (7) above, it appears that the 2nd and third expressions in Lucas (4-44) are incorrect intermediates
  as E0(r) does NOT factor out as shown from E0s(r)*(b_v•r_h)*r_h(r_hb_v).  

/*	From (7), separate terms with E0(r) and E0s(r) within the curly brackets  : 
/$		F(r,v) 
		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*{ E0(r) + β^2*cos(θ´)^2*E0(r)					- E0s(r)*(b_v•rh)*rh(rhb_v)  - (vs/c)^2*E0(r) }      
		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*{ E0(r)*[ 1 + β^2*cos(θ´)^2 - (vs/c)^2 ]	- E0s(r)*(b_v•rh)*rh(rhb_v) }      
/*	subbing vs/c = b 
/$		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*{ E0(r)*[ 1 + β^2 *cos(θ´)^2 - β^2 ]			- E0s(r)*(b_v•rh)*rh(rhb_v) }      
		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*{ E0(r)*[ 1 + β^2*(cos(θ´)^2 - 1)  ]			- E0s(r)*(b_v•rh)*rh(rhb_v) }      
8)		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*{ E0(r)*[ 1 - β^2 *sin(θ´)^2       ]			- E0s(r)*(b_v•rh)*rh(rhb_v) }      
/*	multiply q*(1 - b^2)/(1 - b^2*sin^2(Op))^(3/2) into curly brackets :
/$		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*  E0(r)*[ 1 - β^2 *sin(θ´)^2]	- q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*  E0s(r)*(b_v•rh)*rh(rhb_v)       
		=  q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(1/2)*  E0(r) 								- q*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*  E0s(r)*(b_v•rh)*rh(rhb_v)       
		=  q*E0(r)*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(1/2)									- q*E0s(r)*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)	*(b_v•rh)*rh(rhb_v)       

/*	Summarizing, but writing E0s(r) = |E0(r)| : 
/$9)	F(r,v) 
		=  q* E0(r) *(1 - β^2)/(1 - β^2*sin(θ´)^2)^(1/2) - q*|E0(r)|*(1 - β^2)/(1 - β^2*sin(θ´)^2)^(3/2)*  (b_v•rh)*rh(rhb_v)       


/*++++++++++++++++++++++++++++++++++++++
/*add_eqn  "Lucas_Typo_or_omission   
   04_44rev1 30May2016
   F_electromag_total_constant_v 

/$L		F(r,v) =  q*{ET(r,v) + (v/c)Bi(r,v)}  =  q *E0(r) *(1 - β^2)/(1 - β^2*sin^2(O))^(1/2) - q*|E0(r)|*(1 - β^2)/(1 - β^2*sin^2(O))^(3/2)*(b_v•rh)*rh(rhβ)  
/$H		??????????

/%		FTodv(POIo,t)  =  Q(particle) *E0odv(POIo,t) *(1 - β^2)/(1 - β^2*sin^2(Aθpc(POIo(t),t)))^(1/2) - Q(particle) *E0odv(POIo,t) *(1 - β^2)/(1 - β^2*sin^2(Aθpc(POIo(t),t)))^(3/2) *(beta_v•Roch(POIo))*Roch(POIo)(Roch(POIo)beta_v)]  

/* where   beta_v = beta*Vonv(PART)
	OK - works, Problem - usage of angle Oo in observer reference frame (RFo) instead of Op in (RFp)  
   Oo = Op really applies ONLY when frames are coincident & aligned - here Lucas should use (RFp) primes (eg Op)  
   should Roch(POIo) below be Rodh(Vonv_X_Rpcv(POIo)) from file "Howell - Background math for Lucas Universal Force, Chapter 4.odt"?