380 Prof. P. G. Nutting on the 



then combine them in any manner such that each argument 

 shall enter the function of the other as a parameter, without 

 affecting its form. Consider first the emission as a function 

 of the temperature, the wave-period being a parameter. 

 Experimental evidence indicates that this function is finite 

 and continuous for all periods and for all substances in all 

 conditions, for all values of the argument from zero to 

 positive infinity. The only real root o£ the function is zero. 

 The first derivative of the function is always positive and has 

 no real roots except zero and infinity, nor has it real maxima 

 nor minima, at least when there is no change of chemical 

 phase. Neither the function nor its first or second derivatives 

 has apparently finite, real roots at temperatures of fusion or 

 vaporization. We reject all polygenic and automorphic 

 functions, as well as elliptic and circular functions of real or 

 complex period, for the function is finite and continuous with 

 a derivative, in all finite regions. 

 The inverse exponential 



K T = ae- b < T (1) 



is a simple function satisfying the above conditions. It is of 

 course not the only function satisfying them, but it is probably 

 the simplest in form, and further, it is unrestricted, elastic, and 

 easily meets other conditions to be imposed later on. There 

 are but few algebraic functions whose only real root is zero 

 and whose derivative has no real roots except zero and 

 infinity. A limited class of the form 



E^T^T)//^), (2) 



however, do satisfy the conditions, provided f Y and/ 2 have no 

 real roots, n is greater than unity, and the degree of the 

 numerator is the same as the degree of the denominator. 

 The parameters a and b in (1) may be functions of any argu- 

 ment whatever except temperature. We do not regard them 

 as functions of the temperature, for if we replace either a, b, 

 or T in (1) by/(T), we get an emission which is either not 

 zero at zero of temperature, or else has a finite maximum or 

 minimum. 



The emission as a function of the period of the emitted 

 radiation is a single-valued, continuous function. It has the 

 value zero at zero and infinity, and only at these points. For 

 a black or perfect radiator, it has a single maximum varying 

 with the temperature. Its first derivative has real roots only 

 at zero, infinity, and one finite point. *The inverse exponential 



