22 



For large values of k{a=:s, i.e. 7i = 1) it converges very greatly, 

 and rapidly approaches the first term, i. e. '>iX^:n\^ 



For sjnall values of k (near 0, i. e. a large with respect to s, n := 0) 

 the series becomes : 



n Lè a)^' - yV (7.-^)^ + yio i'/.W - etc.] = 71 {I- cos h Jt)=n. 



For the two limiting cases ?i = 1 and 7i = we, therefore, find 

 back the same values as we had already found by direct integration 

 in the text of ^ 17. 



When n = 0,6, we get 1— 8^'' = 1—4,5 = — 3,5, 1—88/,' + 

 + 136/L'^ = 1—49,5 + 43,0 = —5,5, 1 : (1 + k') = 0,64, so that 

 with V^JT* = 2, 4674, the integral being put =: sn X Vs ^ (^^'- the text 

 of § 17), we find from 



^ 1 ('/,ar 1-8F C/^Jty 1-88P+136^^ (V^rr)^ ^^^ 



*— i^p 12 "^(1-^P)' 360 (1+^'T 20160^^^' 



for t the value 



1—0,1316—0,02425 + 0,00107 . . . = 0,8452 . . . = 0,845 . 



1 

 B. The integral k I -^ tp(/if' (addition to § XVIII). 



In entirely the same way as for the above treated integral we 

 find through repeated partial integration : 



>S; 



tgd, lo,f V e log' 



COS h tiJ 



l/l-f-F cos* At}? 



/3(1 + P) 2 \log 



sec0„ 2 sec^é^. 6 



sec' (9. sec^dj 24 



I/' 





m which (0(j represents iO(j \ ^ " 1/ ~U j' 



In this it has been taken into account that d cosh xp = sink if? and 

 dsinh \p = cosh xp, and that further — P cos^h i|? can again be replaced 

 by 1 — to (when namely 1 + ^'' cosVi if? is put = oj) and — k'' sbfh i|? 

 by — Ic^ cos^h x|, -(- p — (1 + Jc') — oi. Now the terms with odd powers 

 of V' do not disappear, because at the lower limit the factor sinh xp, 

 which occurs for these powers, does not disappear (as for the above 



