SECT. 4] LIGHT 405 



V the volume under consideration. Equation (16) is an equivalent expression 

 in terms of N. When this operation on the observable quantities Nr, Qr, Ni*{d) 

 and l{d) is examined in detail, we uncover the following set of experimental 

 facts : 



(1) a{6) is found to be independent of the amount of irradiation NrQr' 



(2) a{d) is independent of the magnitude of l{d). 



(3) If 6, r, r', I and A^o are all held fixed and G is swung around the beam, 

 (j{d) remains fixed. 



(4) a{d) is independent of the absolute orientation of S and G about p. 



These four experimental findings form the basis for the conclusion that a{d) 

 is an inherent optical property of the medium. This function a, which depends 

 only on 6 (in a homogeneous medium), is called the volume scattering function. 

 Its operational definition is given by (16), or by the equivalent form 



a{d) = N^{d)INQ, (17) 



where 



N^{d) = Ni''{d)llid) (18) 



is a quantity independent of the length l{d) (fact 2). N and Q refer to the 

 radiance and solid angle subtense (at the point p) of the irradiating source. The 

 dimensions of a are per unit length per unit solid angle. Both the unit of 

 length and solid angle are in the direction of observation of the irradiated 

 volume, 



{ii ) The path function N^ 



N^{d), as defined in (18), is interpreted as follows : it is the radiance per unit 

 length in the direction of the line of sight, generated by the scattered light of 

 the beam. 



iV^ is called the path function. It plays an important role in the general 

 theory of radiative transfer and in the solution of visibility problems. By (17) 

 we may write 



N^{d) = a{d)NQ. (19) 



It is easy to generalize this formula to the following form : 



N^ {p, d,ci>)= { a{p, d, 0, e', f ) N{p, d', cj>') dQ{d', cf>'), (20) 



Jin 



where the point p is now being irradiated by flux from all directions about 

 p. An example of the calculation of N^ {p, 9, <^) in real media is given in Preisen- 

 dorfer (1956). a{p, 6, ^, 6', cf)') is the value of the volume scattering function at 

 point p for fight incident in the direction {6', cf)') and scattered off in the direc- 

 tion {6, (f)). {6, cf)) and {6', (f>') are measured with respect to some fixed reference 

 frame [in the derivation {6, 0) was taken as (0, 0) and <y{p, 6, cf), 6', cf)') was 

 conveniently abbreviated to a{d)]. This generalized form N^{p, 6, j>) has the 

 same general interpretation as N^{6) above, but now the radiance per unit 



