473 GENERAL CASE OP THE 



and thence 



, _. nek J ■iak'k" 6u , . 



^"^-^-^^ '{]p^''{i+k'iiUi—k"iiW{{l—u'Ui+ku)Y ^^^ 



These are the simplest forms in which 6t and dv can be 

 presented. It is impossible, however, to express their in- 

 teo^rals in finite terms by means of circular arcs and loga- 

 rithms ; but series exhibiting their approximate values may 

 be found in the following manner : 



Putting, for the sake of brevity, the constant coefficient 



ok 

 — r = ^} and expanding the factor {i-vkii)—^ by the bi- 



g{p—<i) to J J 



nomial theorem, we have 



- -.yx /» dM c ^I l'3i2 2 4.3,5,3 1 4.3.5.7,, X o -i 



in which, if we put the integral of — r- jr=arc(sin.=a;)=^, 



d?/ 

 V(l^')' 



and the integral of _ ., = — \/{i—x)=M^ we shall have 



7/'d7/ J\—Us/Y—oe^ B 



f 



J VI— m' S.4 ~ 2.4' 



J VI— m'~ S.4.6 ^3.4.6' 



^C. «JC. 



u'Au _ 3JB— az/yi— m" C 



U^du 5C— 3.4MV1— W^ -O 



