404 TYLER AND PREISKNDORFER [CHAP. 8 



go through the details of this computation because they lead us directly to (i) 

 the volume scattering function, g, (ii) the path function, iV^, {in) the total 

 scattering coefficient, s, and {iv) a simple derivation of the basic equation of 

 transfer for radiance. 



(?) The volume scattering function 



Assume r' is small, so that the surface radiance of the small segment of the 

 beam under view by G is essentially the recorded field radiance, Ni*{9). Let 

 A{6) he the projected area that the observed element of volume presents to 

 the line of sight of G. Then by (1), the flux per unit solid angle emitted by the 

 volume in the direction of G is clearly Ni*{6) A{6). But the volume of the 

 observed element of beam can be represented by A{d) l{d), so that the flux in a 

 unit solid angle in the direction of G emitted by a iniit volume of the mediimi 

 at p is evidently 



Ni^d)Aid) ^ Ni^d) 



A{d)l{9) 1(6) ^ ' 



Let the cross-sectional area of the beam be designated by A. Then the quantity 

 ANi*{d)ll{9) has the following simple interpretation : it is first of all the amount 

 of flux ])er unit sofid angle scattered in the direction of G ; and secondly, it is 

 scattered by an element of volume of the medium which has cross-section A 

 and imit length in the direction of the beam. Thus the integral 



N!*(9) 



over all solid angles about p evidently gives the total flux scattered out of the 

 beam per unit length of travel of the beam through the medium. 



Equation (14) is useful in practical estimates of the rate of loss of radiant 

 flux from the beam through the mechanism of scattering ; it also contains the 

 germ of the idea of the volume scattering function. To see this, we first note 

 that the total flji.r of the beam across the area .4 is NfQrA , where Qr is the solid 

 angle subtense of the source at point p and Nr is the radiance of the beam at r. 

 Therefore, if we divide (14) by this quantity, the result, 



iV/*(^) 



has the following interpretation : it is the total amount of radiance lost by 

 scattering per unit length of travel of a beam of unit radiance. 



We now may inquire about the directional distribution of the scattered flux. 

 From (15) we see that this distribution is governed by 



The definition of a{6) is sometimes written a{d)=^J{6)IHV, where J{6) is the 

 intensity of the scattered light in the direction 6, H is the irradiance input and 



