412 TYLER AND PREISENDORFEB [CHAP. 8 



The right-hand sides of all these inequalities hold without qualification. 

 However, the left-hand side of (42) holds whenever O^K{Z, + ). The left-hand 

 side of (44) holds whenever K{Z, -)^0. The condition O^K{Z,+) almost 

 always holds, so that the inequalities of (42) for down welling streams almost 

 always hold. However, the condition K{Z, — ) ^ hardly ever holds, so that 

 the left side of (44) for the upwelling stream hardly ever holds. The condition 

 K{Z, — )^0 means that the down welling stream is constant or growing with 

 increasing depth, a situation which occurs, if at all, only in regions of very 

 shallow depths in the hydrosol, or in regions where there are self-luminous 

 sources distributed throughout some layer. 



Some further inequalities which are helpful in checking experimentally 

 obtained optical properties and also aid in the understanding of the mutual 

 interactions between the up- and downwelling stream of radiant flux are : 



K{Z,+)R{Z,-) ^ K{Z,-) (47) 



or, equivalently, 



dHjZ,-) dH{Z,+) 



— IT- ^ —Jz — (^^) 



These relations hold for arbitrarily stratified source -free media. The same is 

 true for : 



^^^^ = R{Z, - )[K{Z, - ) - K{Z, + )]. (49) 



The quantities a{Z, ± ), a{Z, ± ) defined in (41) and (46) are hybrid optical 

 properties : they are the result of simple combinations of the inherent and 

 apparent optical properties. Equation (41) gives the volume absorption func- 

 tion for each stream, and (46) gives the volume attenuation function for each 

 stream. These quantities by definition do not fall directly into either the 

 inherent or apparent class. 



To round off and complete the picture of the hybrid optical properties, we 

 mention the {volume) forward scattering functions : 



f{Z, ± ) (50) 



the {volume) backward scattering functions : 



b{Z,±) (51) 



and the {volume) total scattering functions : 



s{Z, ± ) (52) 



for each stream. Detailed definitions and discussions of these quantities may be 

 found in the references (Preisendorfer, 1957a). The hybrid optical properties 

 play important roles in the exact theoretical discussions of the two-flow analysis 

 of the light fields. They are also of use in collating experimental data on in- 

 herent and apparent optical properties. Examples of such uses may be found 

 in the references (Preisendorfer, 1958). 



