24 



side of the first term, and besides the whole part vvitli k\/<p disap- 

 pears. That in tliis case oiilj the first term with /oy"' remains, follows 

 also from this that tg 0^ =^ k"" {\-[-fp) :{l~k''<p) approaches h for 

 <p=.0, so tfiat in case óf equality of the limits of the original integral 

 the factor k cos h\p ■.\^l-\-k'' cos^li^]^ =^ k :y^l-\-k'' does not change 

 between them (with respect to the log that becomes at both the 

 limits), and can accordingly be brought outside the integral sign. 

 At low temperatures (but higher thafi the limiting temperature 



T„ determined by -,. = ]: k') the whole second part of kj will 



again disappear in consequence of the factor 1—^V^, which approaches 

 0, whereas of the tirst part again only the first term with log" remains. 

 In this case cos h xp =^ tg 0^ : /• = oo at the lower limit, and the 

 factor of if?f/i|? in the integral can again be placed outside the integral 

 sign at this limit, which now prevails since the log l)ecomes infinite 

 there. At the other limit the log is namely = 0. 



With close approximation we may, therefore, write [a has been 



written for k : V^l^V = s:a): 



J ■ ' l/l— y^V 



with neglect of all the terms with higher powers of log. Only at 

 intermediary temperatures the omitted part can have any influence 

 — but the difTference brought about by this might possii)ly be made 

 to disappear entirely on a somewhat modified assumption concerning 

 ƒ(?■) between a and s (see § XVI). 



C. The quantity a for 7 = r/^ = 1 : /(.-. (addition to ^ XX). 

 The original integral was (cf § 16): 



2a' rr r{-f\r))drXsinOdd 



2a' rr 



a = 4 X (6,). « X ^^^ jj 



^l-'^sin0 + rpf{r) 

 We may also write for the integral: 



o 



rr r'i— f'{r))dr diacosO) 1 /' ,...,,/'« 1 /. , «'A 



J J \/r\ff{r)—{a'—r') + a'cos'd «J \'ï ^ 9/ 



when r*(ff{r) — {a' — r'') = (f is put. When f {r) is generally 



= -, so that this duly becomes =1 for v = 5, then — ƒ'(/-) = 



and r/^ = (a^ — /'^). Hence we now have: 



