nutting: gray radiation 477 



hence the necessary and sufficient condition that the pencil 

 of lines represented by (4) be stigmatic is that a be a linear func- 

 tion of b, 



a = ijo + bx (6) 



x and y being constants independent of both wave length and 

 temperature. 



This condition however is not satisfied even for equilibrium 

 radiation, for a = log {Bi A -5 ) and b = B*/\, and neither of 

 these expressions can be a linear function of the other. Over 

 but a moderate range of wave lengths the expression holds to 

 a fair approximation (probably to within the limits of experimen- 

 tal error). For, let X = X (1 =•= 5) where 8 is so small that its 

 square may be neglected in comparison with its first power. 

 In this case a and b are both linear functions of 8 and hence of 

 one another. 



Consider now the free radiation represented by (2) whether 

 gray or selective. The parameters C h n and C 2 vary not only 

 from surface to surface but (in general) with both wave length 

 and temperature; in other words, the equation is too simple 

 to represent free radiation. However, the stigmatic condition 

 may be applied even though no parameters are constant. If 

 the data indicate that a number of logarithmic isochromatic 

 lines pass through a common point, then an equation similar 

 to (6) must hold over the range of wave lengths covered by the 

 data. Hence, for any variation da = x db; for example, da/d\ 

 = x db/dX. The linear relation (6) requires 



log CiX— = 7/o + — - (7) 



A 



Hence, by substitution in (2), in any region within which 

 the stigmatic condition holds, even though d, n and C 2 vary, 



E CJl 1\ 



^ Er i:\rr t) (8) 



where E and T are fixed constants such that log E and 1/T 

 are the coordinates of the point of stigmatism. 



This is of the same form as the Paschen-Wanner equation, 

 used so much in monochromatic pyrometry, but with a some- 



