Manchester Memoirs, Vol. lix. (191 5), No. 13. 17 



Fig. 11 shows the method of construction. A OB is 

 the angle to be trisected. is the origin and OB the 

 axis of x. OC\s made unit length, 0F=0D = 2\0C) = 2. 



0S=~ , <9£ = -^ 2*^= #2 - OFsin ADB=2 sin 3a = 2a. 

 16 8 ^ 



The parabola has its vertex at and focus at 5. The 

 circle has its centre at H and passes through O. PQ is 

 drawn parallel to OB through the intersection P, and OQ 

 is made equal to OC; this is unit length. Then if the 

 ordinate of P is ;', 



sin 00 '6 = — -= = y = sin a = sin . 



OQ 3 



Hence the angle QOC is one-third of the angle A OB. 



(4) Oilier Methods derived fi om those already given. 



In any of the preceding cases (IV. I, 2 or 3) the 



equations of the circle and conic may be combined to 



form an equation of the second degree, which will be the 



equation of another conic intersecting the circle in the 



four points common to the circle and the original conic. 



For example : in the method of Clairaut (IV, 2, Fig. 10), 



if H be taken as the origin and BH as the axis of x, the 



equations of the circle and hyperbola will be : 



x- + ?.ax +y- + 2by = 0, 

 and 



3# 2 + 4a x -y = 0, 

 where 



a = EH and b = SF. 



Hence, if X be any constant, the conic 



\(x- + zax +>- + 2by) + 3.V- + ^ax -jr = 



will cut the circle at the four points //, P, P xi J\,, and 



may therefore be used to trisect the angle BSH. Any 



number of solutions are possible, since any value may be 



assigned to X. Some of the simpler cases arc as follows : 



