Lucas "...   using a spherical surface centered about the charge 
	distribution q with spherical coordinates and noting that E and n 
	will then be in the same direction   ..."

/$			4*π*q  =  ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0(r)*r*sin(θ)*(1 - λ(v))/(1 - β^2*sin(θ)^2)^(3/2))) 
		=  4*π*q*(1 - λ(v))/(1 - β^2)  

/$			4*π*q  =  ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0(r)*r*sin(θ)*(1 - λ(v))/(1 - β^2*sin(θ)^2)^(3/2))) 
		=  4*π*q*(1 - λ(v))/(1 - β^2)  


/*	Howell - Collecting Lucas's result for (4-39) and (4-40)
	04_39 E(r,v) for constant velocity, non-point charge, observer reference frame  
/$		ET(r,v) = (1 - λ(v))*E0(r)/(1 - β^2*sin(θ)^2)^(3/2)  
/*	04_40 L(v) expression for Gauss law for electric charge 
/$		4*π*q  = ∬[∂(Area): ET(r)nh)   

/*  Expressing (4-40) in terms of constant r, variable [O,P] to integrate
  .over.the spherical surface : 

/$1)			4*π*q 
			= ∬[∂(Area): ET(r)nh)   
			= ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: ET(r)nh)

/*	Subbing (4-39) into (1) - BUT Lucas has extra "r*sinO" term
***Where did that extra term come from? - may be a hint for (4-37) etc!!!!

/$		  4*π*q 
		= ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: (1 - λ(v))*E0(r)/(1 - β^2*sin(θ)^2)^(3/2)))   
		= (1 - λ(v))*E0(r)		*∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: (1 - β^2*sin(θ)^2)^( - 3/2)))   
/*  As there is no change with P : 
/$		= (1 - λ(v))*E0(r)		*∫[∂(φ),0 to 2*π: φ)*∫[∂(θ),0 to π: (1 - β^2*sin(θ)^2)^( - 3/2)))   
1a)	= (1 - λ(v))*E0(r)*2*π	*∫[∂(θ),0 to π: (1 - β^2*sin(θ)^2)^( - 3/2)))   

/*	To assess the integral in (1a), expand the series as per (4-38)
/$		(1 - β^2*sin(θ)^2)^(-3/2) = (1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 + 35/16*β^6*sin(θ)^6 + ...)  
/*	So 
/$			∫[∂(θ),0 to π: (1 - β^2*sin(θ)^2)^( - 3/2)))  
		=  ∫[∂(θ),0 to π: (1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 + 35/16*β^6*sin(θ)^6 + ...)
		=  ∫[∂(θ),0 to π: 1)  
			∫[∂(θ),0 to π: 3/2*β^2*sin(θ)^2)  
			∫[∂(θ),0 to π: 15/8*β^4*sin(θ)^4)  
			∫[∂(θ),0 to π: 35/16*β^6*sin(θ)^6)  
			∫[∂(θ),0 to π:  + ...)  
/*	Ignore the last term, integrate the others 
     =  π  
/$1a1)		 3/2 *β^2*∫[∂(θ),0 to π: sin(θ)^2)  
1a2)		15/8 *β^4*∫[∂(θ),0 to π: sin(θ)^4)  
1a3)		35/16*β^6*∫[∂(θ),0 to π: sin(θ)^6)  
/*	Using Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt
	Recurring integrals from http://integral-table.com
/$1a1a)	∫[∂(θ),0 to Of: sin(θ)^2)   =  O/2 - sin(2*O)/4  
1a2a)		∫[∂(θ),0 to Of: sin^3(O))   = -3*cos(θ)/4 + cos(3*O)/12   

/*  This is NOT going to work!!  From where did Lucas get the extra "r*sinO" term?


/*	Now looking at 
/$			∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: ET*(1 - λ(v))*r*sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]}   
/*	[ET, L(v), b, r] are pseudo-constants wrt dO, so they come out, note above Lucas error with integral limits, but now ET instead of E0!
	Also Lucas missing a second r for dP !!???!!
	Correcting original : 
/$			∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0*(1 - λ(v))*r*sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]*r}   
			∫[∂(φ),0 to 2*π: E0*(1 - λ(v))*r^2*∫[∂(θ),0 to π: sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]}   

/*	Now looking at 
/$			∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: ET*(1 - λ(v))*r*sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]}   
/*	[ET, L(v), b, r] are pseudo-constants wrt dO, so they come out, note above Lucas error with integral limits, but now ET instead of E0!
	Also Lucas missing a second r for dP !!???!!
	Correcting original : 
/$			∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0*(1 - λ(v))*r*sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]*r}   
			∫[∂(φ),0 to 2*π: E0*(1 - λ(v))*r^2*∫[∂(θ),0 to π: sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]}   
/*	using result above for ∫[dO, 0 to π : sinO/(1 - b^2*sin^2(O))^(3/2) }
/$		=  ∫[∂(φ),0 to 2*π: E0*(1 - λ(v))*r^2*2/(1 - β^2)}   
/*	E0s has constant scalar magnitude at constant r for integral 
/$		=  E0*(1 - λ(v))*r^2 *2/(1 - β^2) *∫[∂(φ),0 to 2*π: 1}   
		=  E0*(1 - λ(v))*r^2 *2/(1 - β^2) *2*π 
/* sub  E0 = q/r^2
/$		=  4*π*Q/r^2*(1 - λ(v))*r^2/(1 - β^2)
		=  4*π*Q*(1 - λ(v))/(1 - β^2)


/*++++++++++++++++++++++++++++++++++++++
/*add_eqn  "Howell_incomplete   
   04_41
   L(v) expression for Gauss law for electric charge  

/$			4*π*q  
		=  ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0(r)*r*sin(θ)*(1 - λ(v))/(1 - β^2*sin(θ)^2)^(3/2))) 
		=  4*π*q*(1 - λ(v))/(1 - β^2)  

/*	havent done yet 
   PRIORITY PROBLEM - sin,cos terms