384 Prof. P. G. Nutting on the 



Maxima of types II. and III. may be introduced into 

 functions (8) and (9) by simply applying to them an inverse 

 algebraic polynomial in r as a factor. The real roots of this 

 factor would give infinite maxima, independent of the tempe- 

 rature of type III. And near each imaginary root would be 

 a real maximum, finite and rather broad, of type II. But 

 there are a number of conditions to be observed in the insertion 

 of these maxima. In the first place, the emission-temperature 

 function, which determines the isochromatic curves, must 

 remain unaltered in form. Again, if all bodies emit like 

 perfect radiators at low temperatures and long periods, and if 

 these period-maxima are all of short period and occur at high 

 temperatures, then the complete emission-function must return 

 to the original form (8) or (9j for perfect radiators when we 

 make these maxima zero in the function. And when we place 

 these maxima equal to zero in the function, the form of the 

 function must not depend upon the number of these maxima ; 

 so that the polar factor in the complete function must consist 

 of a summation of simple factors of similar form, rather than 

 a polynomial of high degree. 



After unsuccessful trial of many possible polar factors, the 

 form 



was found to be satisfactory. This factor assumes that there 

 are p maxima of type II. and q maxima of type III. We have 

 then for our complete function, instead of (8) and (9), 



m n 



E=Tt-vW-2 -jJ* (10) 



and 



E = ATVtfTT) 2 -^ .... (11) 



in which we have written as one the two terms of the polar 

 factor above and <f> = </>i/$2- The logarithmic derivatives of 

 (10) and (11) give 



_4t 2 



t 2 ±t; 



and 



4t 2 



2 t-,-¥ T + « ...... (12) 



t" + t< 



F(tT)-f, (13) 



considering but a single maximum of each type. In (13) we 

 have written P(tT) for t.0 / (tT)/^>(tT) j remembering that 



