10 



1-. 



is = s'' : a\ The limits for .'■ are 1 and /;, hence 1 / ^ 



, / 1 ^ » 



and for //, i. e. for tcjhxp. Thus / ^ and 1 for co.yA il^= 



= i : \/a—tfhxp), or t(/ 0, : /,■ and 1 ; f—x,"- being = k\v,\ and .r„^ 



being cc.y" ^>,. Evidently the limits for if? are % 





= A.(^.l/^-l)a„aO, 



as 



BgUjhM;>— - /09 ; — — loq " ^ i-. 



In this hjO^ : k is > 1, because now p <^ I. 



Thus we obtain an integral of quite the same form as that of 

 ^ XVII; with only this difference, that now hyperbolical cosinus is 

 put instead of the former sinus. When again we expand into a 

 series (see Appendix B), it appears that both at high temperatures 

 {(p==0) and at loiv temperature (rp = y„ = j:p) all the terms with 

 higher powers of log with respect to the first term disappear, so 

 that with close approximation we may write-. 



— ^nVl -\- fp log'' , 



\/\—k'fp 



in which (p is determined by the relation — siti^ 0^ =z i -\- (p (cf 



equation (6) of the previous paper), in consequence of which «"//V^ : /," 

 becomes = (1 -|- (p) -. (1 — k^'q). {n has again been written for 



When we now add the found integral to the first, viz. ^ %' 



1 + y^l-p' 



P 



a' 



then (// being = ^ .t-/ = (l-j-p).,/, and .^•„* being = 1 — slnVJ^ 



a — s 



= 1 — - — — (1 +7^), so that p^ becomes = 1 — k'' (p) we get: 



1 -p A^" 



Za 



so that taking Into account the factor o) X{2 a* : s{a'^ — ó'")), we get 

 the following form : 



