102 DIFFERENTIAL AND INTEGRAL CALCULUS. 



or = sin>Jr = sin ^r on the same branch as before, 



X ~f" Z(t 



when (x = a, ty \ir, 



2a sim/r 2a 



1 sin\/r 1 sin^r 



dx 2a cosilr cc c?5 , ds 



therefore -y = 7- . . N!4 = -7 . -j-r = cos-yr 



-' r 



therefore -77 = 7= . 



d\jr (I - si 



therefore s = 2a 

 if = TT + r or 



=a [(1 



J 



f 9 



-. 5 = 2a T-- 

 1-sin^r)" J o (l+ 



Z COS 



<9 + tan :! 



T - ^ - A 



1-fcos^y 3 l+cos0 



cos ^(2- sini/r) 



Now 



1- _^ 

 sin 



2 

 therefore MK = MP cot i|r = - 



_2a //l + sini//\ 2sini//-l 

 = "3" V U-sin^J ' 1-sTm/T 



_ 2a cos^/ (2 sin i// 1) ^ 



therefore OK= . ^ C . 8 '* , {3 (1 - sin;/,) -f (2 sun/,- 1)} 

 oil bin \jj j 



<2a cos\L (2 sin*/,) 

 = - ^ v . r/ = K 



3(1 -sm^) 2 



