322 CLAUSIUS ON THE CONSTITUENTS OF THE ATMOSPHERE 



this purpose, we will set g dg da instead of day, and then in- 



TT , TT 



TT TT 



tegrate from a = — — to a= — , and from ^=0 to g=2{i—i') ; 



we thus obtain 



We can here set sin^ («— i') for (i— ^')^ and also simplify as we 

 did before in formula (11) ; we then obtain 



J « • • •^- • 2/- -/xfi „sin2(i— i') sin'*(i— i')1 ._ . 



<^y.2sm^cos^c?^sm2(^— ^' 1 — 2 . ^]. , .,; + . 4 . , ., . . (16«) 



^ ^ T sm^(^^-^') sm^(^^-^')J ^ ' 



This expression, which represents the intensity which passes 

 through dy, taking merely the dispersion of one elementary 

 zone into account, must, to find the total intensity which passes 



through dy, be integrated from i=0 to i= -. 

 The integral thus determined is 



'^y'~w~ L 8 ^^^w-i 12 



5{n^\f "^ 42(7^4-l)'* J' ' ^ '^ 



and, for the sake of brevity, let us denote it by dy , N. Now as 

 the same quantity, according to the former notation, was ex- 

 pressed by ka dy, we obtain 



N 



^= ^ (17«) 



We have therefore found two separate expressions for h and k, 



(1 2«) and (1 7«) • In these, indeed, an unknown quantity a- makes 



k 

 its appearance ; forming however the fraction rr, which it was our 



object to ascertain, the said quantity disappears, and we have 

 remaining simply a function of the index of refraction n, namely, 



\'l M 



To obtain definite numerical values from this formula, we 

 must make assumptions as to the possible value of n. Among 

 the indices which can enter our considerations here, the fol- 

 lowing may be mentioned. From the experiments of Dulong 

 we find. 



