1 1 



11 /' dx A- J/V-p" r dx 



We liiive written — dx for sin OdO =. — dcosO. The minus sign 

 has again been removed by reversing- the limits of integration. 

 Besides — for the saice of homogeneity — a factor /; has still been 

 introduced under both lo(j. For slap we may also write v p^ — .';„*• 



The first integral can again be easily integrated, d loc) is namely 



dx 

 = ; , SO that we tind for it : 



, /x—}/x^^p'\p , 1—1/ r^"^ , p 



h %i ^ ~ = - i %' -= - ' %"^ r7T7-= ' 



V p /I P 1 + |/l -p' 



1-fi/n^' , 



for which with a view to log^ also — h ^og' ^^^y "^ 



written. 



The second presents again the same difficulties as the correspond- 

 ing B(/t(/ in § XVII. This becomes namely, d log now being =: 



X dx 



\/a' — x,' S/x^'—f 

 1 



rv^x^—x,-" r ^p'—x^' , ( /I— V 



— ~ log X dlog= — — %/ / 7^ X « ^og, 



J A- ./ l/p^-r/\f;^ ^ 1 + y 



seeing that 



while from (.6'"—//) : (.<;' — .«,") = ƒ follows .i-' = (// — i/'-^'o^) = (1 — .V') 



and .6'»— .i"/ = (/j'— .t'o') : (1--^"). Now loq I / ~ = —Bqtqhyp y, 



' V l+y 



so that we tind with Bg tg li y =^ \\> -. 



V 1 



-^ ^ ^ dx\y = k ^ — = if^^/ifN • 



P ^f y .vj J 1/ 1 + F cos Vi tb 



1 '.tgVi^j u/da-.k 



P 



because \- - cos^i ip can be substituted for co.s-vi ifj — X 



a' a" rt" 



X A"m*A t|j, with siirh if> = C(w"A if — 1 ; and .v' : (a' — cv') is = k^ (see § XVI I). 

 For A'o" : //' vve may namely write {a" — .v') : a', and (;>^ — 't'o') = /^' 



