406 TYLER AND PREISENDORFER [CHAP. 8 



length in the direction of observation is generated by hght scattered into the 

 line of sight from all directions about the point p. By property (4), 

 a{p, 6, (f>, 6', cf)') for any pair {6, cf)), {9', (j)') is known from the determination of 

 a{d) at point p, as defined in (17). 



{in ) The volume total scattering coefficient 



From the above arguments we now have an explicit expression for the term, 

 s, which arose in the discussion of the volume attenuation coefficient. For this 

 term is evidently none other than that given in (15) which, by (16), may be 

 written 



s = ( a{d) dQ = 277 f" a{d) sin 6 dO. (21) 



J47r jo 



The second expression follows from facts (3) and (4). This is the volume total 

 scattering coefficient. In non-homogeneous media, it may change from point to 

 point but, in any event, s does not depend on the direction of the irradiating 

 beam [facts (3) and (4)]. Closely related to s are the {volume) forward scattering 

 and {volume) backward scattering coefficients, f and b respectively, defined by 

 the following formulas : 



/ = 277 \ ' a{d) sin d dd, (22) 



jo 



6 = 277 f " a{d) sin d dd, (23) 



Jn/2 

 so that 



s=f+b. (24) 



As in the case of s, both / and b may vary with position, but they do not in 

 any event depend on the direction of the irradiating beam. 



{iv) Equation of transfer 



From the preceding interpretations of a and N^ , it is easy to verify that the 

 equation of transfer for field radiance (or surface radiance), Nr, in a source-free 

 medium is expressible as : 



^ = -aNr + N^. (25) 



The first term on the right gives the space rate of loss of Nr by attenuation ; 

 the second term gives the space rate of gain of Nr by scattering. 



c. The volume absorption coefficient 



During the discussion of the volume attenuation coefficient a, we found that 

 a included two distinct types of action by the medium on the beam — absorp- 

 tion and scattering. Thus, if a and s are known, we may obtain a by subtraction : 



a — s = a. (26) 



