Complete Emission Function. 381 



is probably the simplest and most general function satisfying 

 these conditions. In this function, </>(r) must have no real 

 root unless it be zero, and its derivative none except perhaps 

 zero and infinity. A class of algebraic functions of the form 



Er-tfl-ftM/^T) (4) 



satisfy the above conditions, provided <£, and <f> 2 have no 

 real roots and the denominator is of higher degree than the 

 numerator. 



For perfect radiators we have then for the complete 

 function, combining (1) and (3), 



*= A W) e ~ Bh *' (5) 



or, combining (2) and (4), 



E=ATV/ 1 (T)&(t)// 3 (T)&(t). . . (6) 



The combination of (2) with (3) does not satisfy the con- 

 ditions imposed upon (1) and (2). 



Another property of the emission from a perfect radiator 

 shown by a number of investigators* to hold well experi- 

 mentally is that the product of the absolute temperature and 

 the period of the emission maximum is a constant. Mathe- 

 matically this means that the derivative of the emission 

 function with respect to the period is expressible as a iunction 

 of the product period times temperature. This condition 

 rules out the combination (l)-(4), determines <£(t) in (5) to 

 be T n , and simplifies (6) to 



E=AT"Ty(T# 1 (TT)/fc 1 (TT). ... (7) 



Paschen and Lummer and Pringsheim (/. c.) found also 

 that the emission corresponding to the period of maximum 

 emission was proportional to a power of the absolute tempe- 

 rature. The condition limits J(T) in (7) to a simple power 

 of T which may be included in the factor T". 



Provided then there are no maxima independent of the 

 temperature in the emission spectrum of any body, any 

 formula representing the emission is probably included in 

 one of the two following forms : — 



E = At"V- B ' vt , (8) 



E=ATv££J (9) 



* Cf. Paschen. Sifz. K.A. W. Berlin, April 27, 1899 ; Lummer k 

 Pringsheim, Verb. d. D. Ph. Gcs. i. p. 227 (1899) ; Paschen, Ann. Physik, 

 iv. p. 294 (1901). 



