20 

 APPENDIX. 



ƒ sin \p 

 — ifv/if?. (addition to § XVII). 



yl4-k^ sin it' 



When we expand this into a series through repeated partial 

 integration, we get: 



2 Q dip 24dxp^ \20di^y' 



in which (through if') all the terms at the lower limit disappear. 



And for the npper limit all the odd ditïerential qnotients of P will 



disappear, because in (his cosily occurs as factor. Indeed, when we 



dxo o 7 • 



p„t '{A.l-^ sin \V = w, so (hat becomes = 2 A,' .svnipct».s'i|% we have: 



^ ' dip 



dP sitnp , cos if» f—k' sin'' if) 1 



^^ — h—^ {2k' sin ip CO. If') + — p = cos ip f ^7 1 = 



d\p ' to ■/•-.• ^''' V ^'- 



Z=Z COS if' 



CO '/a^ 

 1 O) 1 \ cos%p 



(oV. '^ tovj lo'l^. 



d'P 

 d^ 



cos If) , sin If) . / 



^ (2^' sin If) cos I})) — - =: — sin if) 



3P fos" if) 1 



— sin if' 



3(1+F) 2 



V. 



''/W 



because /-^ nw^if' = k'—k'sin' if =/,•''— (to— 1) = (1+^-')— w. We liave 

 further: 



3(1 +^-') 2 



d*P 



rfif)" 



,v/7< ir I -^, )(— Fsmif'co.sif') — i-o.^if^ 



'V2 to"/! 



I=z — COS if' 



= — cos if' 



And also : 



(O li CO /a 



15(1 + P) 6 



3(1+F) 2 



CO'.'a 0> ■*/'•! y "■ ' ' V <oVa W^/a 



15(1^P) 12(1+P) 4 6 4 



TT 0-to) + 



Vï 



+ 



d}p* 



CO ri 



-\- sin if) 



to'/i ' wVs^ 



15(l + /;0 12.(l + F)4-6 , 4 



CuVi 



toVa 



+ ■ 



CO 'a 



