DIFFERENTIAL AND INTEGRAL CALCULUS. 45 



du ~ dc ~. dc 



therefore -j- = - p -r- -f p -7- + a - a. 



o?a da da 



dc 

 = (/3' - /3) -7- = (/3 1 - 0) x radius of curvature at P, 



75} /JQ> JO\ 



a u i dp dp\ i r>, n 



rr> = r- 1 Pi when 8 = 8, 



da. \da da.J r 



and -. i, - = curvature at = =-y-(l -j-} : 

 d(c a) i\ dc \ den J 7 



, d/3 p d/3' p 



therefore 1 -y- = , 1 4- = , 



1 2 

 or 



therefore for a maximum 



/3 / = ^, and f- - 



112 

 For a minimum /3' = /3. and h - > - . 



(4) M = i [ c 2 y5l 2 ^, where c/(a) = a, 



^0 



,1 /* // / \ ci dc 



therefore / (a) = -- 5 -7- 



therefore = c " y t?0 + 



2 - ^ = ic^f - 2w - /' (a). 

 c t/a 2 ^ J a" 7 - ' 



Now c/(a) = a, nd' 1 = cot OPT (fig. 10) ; therefore for 



t/ \ t 



a maximum or minimum 



= 4 cot OPT 7 or u = 



