220 



Transactions. 



The equation of EG is 



X — a cos I a 



(a - ^J _ cos (a - ^) + CO. (a + ^) 



y 

 which reduces to 



a sin (a - -^ J sin ( a - ^ J + sin (a + ^ J 



a cos I a — - 1 



= cot a. 



•(2) 



?/ — a S! 



These meet on OK at K. Hence put y = and equate the resulting 

 values of x. We get 



cos a — sin a cot (a — -J = cos (a - - j — sin (a - -) cot a 

 wliieh becomes 



sin - 

 8 



sin 



sin 



siu a 



2 sin - cos - 



8 8 



sin a 



.■. sm a 



=. 9 



ji sni 



(•-0 



— a) COS - = sm ( - — al — sm a 



.-. 2 sm a = sin (^ - a\ (3) 



Now, OK being almost coincident with a true trisector, we have a nearly 

 equal to (l --^j — i.e., to 4-- This may also be seen from (3). Hence put 



a = ± 



1-2 



where e is very small. Then (3) gives 



2sin(A-.)=sin(| + .) 

 Expanding and retaining only the first power of e. we get 



2 sin A — 2 e cos -^ = sin - + c cos ^ 



12 12 6 ' 6 



and therefore 



2 sin 



12 



sin 







2 cos _ + cos 

 12 6 



Here i and -i- are proper fractions. Expanding in powers of 0, and retain- 

 ing only the lowest power of $ that remains, we get 



- _^ 

 ^ ~ 5184 



This is the angle between a trisector as given by the construction and 

 a real trisector. The accuracy of the construction increases rapidly as 

 the angle diminishes, the error being approximately proportional to the 

 cube of the angle. 



If the given angle be a right angle the error is 0'067 of a degree, for 

 an angle of 45° it is 008 of a degree, and for one of 30° it is only 0002 

 of a degree. 



In conclusion, we shall indicate briefly how the trisection of any angle 

 between 0° and 180° can be derived from that of an angle not greater 



