for the Measurement of Visibility of Objects. 109 



as asymptotes the lines y = -f 50 and x = ; while 



T> 



Case II., ~ <k, 



I>2 



gives a family of hyperbolas having as asymptotes the lines 

 y = — 51 and # = 0. 



It will be se^n that considerable portions of these curves 

 lie in the fourth quadrant where values of V 6 are negative. 

 It is obviously impossible to have values of visibility less 

 than zero, hence all such values may be considered as unreal 

 or imaginary. The only portions of the curves of interest in 

 this problem are those lying in the first quadrant. It will be 

 noted also that in general visibility is high for very low 

 values of W, decreasing to zero at a value of W which 

 depends upon the assumed value of R 2 , ar *d then rising 

 (approaching V=-+-50 as a limit) for high values of W. 

 It will be noted that for any given value of R 2 there is a 

 small range of W values for which W = 0. If we express 

 this range by the symbol /\W its value may be expressed 

 by the equation, 



AW=E 2 - r . 



Thus the range of weather conditions for which V can be 

 zero is a f miction of both R 2 and k. It will be seen by 

 examination of the curves that V=0 when R 2 =W. An 

 object becomes invisible against a given background when 

 the reflexion factor, R 2 , of that object is equal to the ratio 

 of background brightness, B 1? to the illumination, E 2 , on the 

 object plane, that is when B 1 = B 2 . 



Now, passing on to a consideration of visibility as a func- 

 tion of the reflexion factor, R 2 , as a variable quantity and 

 W as constant, the curves shown in fig. 4 are obtained by 

 solution of equations (11) and (12) for various fixed values 

 of W and the same value of k (k — 1'02) as was used 

 previously. It will be seen that 



CaseL, w l >k, 



B 



is represented by a series of straight lines, all passing- 

 through the point # = 0, ?/=+50. The slope of the line 

 for any particular assumed value of W is given by the 

 expression 



