20 GENERAL THEOREMS, CHIEFLY POEISMS, 



fore, p M : - x AK AQ :: ABX 2EF : A B 2 A c ; 

 M + N 



M ' 



or y : - x arq 90 arc vers. sin. x : : a b : a 2 c ; 

 M + N 



therefore y (a 2 c) or + y (2 c a) = a 6 x ( - -X arc 



\M -\- N 



. 90 arc v. s. x J, and by transposition ab x arc . v. s. x 

 + y (2c a) = - -x arc 90. To these equals add 2y 



M -j- N 



(x x) = 0, and multiply by 1 ; then will a b x arc v. s. 



a b arc 90, of whi 

 ab x arc v. s. x 



x + (2 x a) y 2 y (x c) = - x a b arc 90, of which 



,, .,, 



the 4th parts are also equal; therefore 



(2 x a) y y . . ab M 



- -- - - (x - c) = x - X arc 90. Now be- 



4 2 4 M + N 



IT 9 b* , _ b 



cause A F B is an ellipse, y ^ x (a x x ), ana y = - 



a a 



. f a b x arc v. s. x 2 x a 



V (a x or); therefore - - -- -\ --- - x 



- V (ax x*} - | (x - c) = x - X arc 90. Mul- 

 a 2 ^ 4 M + N 



tiply both numerator and denominator of the first and last 



, ,, & a s 2 x a b 



terms by a ; then will - X -7- X arc v. s. x -\ -- X - 



a 4 4 a 



?/ & (Z^ M 



*/ (ax a?) ?-(x c) = -X-rX -- X arc 90. Now 

 Z a 4 M -|- N 



the differential of an arc whose versed sine is x and radius , 



2t 



ft fjt nn __ 



is equal to . , - , which is also the differential of the 

 2 sj (ax or) 



x b /a 8 



arc whose sine is V~ and radius unity; therefore- x I X arc 

 a a V 4 



