OPERATIVE ROOTS OF THE CIRCLE-FUNCTION 129 



Kb. 



; so that 



circle holds regarding the similar chords of any other circle. If a 

 then ma = K^^nib ; so that 



KlfnK^K^ . . . = KlKiKlmO . . .; 

 mABC . . . = AfnBmCmfn . . . ; 



AmfnO = mA. 



_ ma mh mc _ mo my my 

 000 000' 



and 



or 



and conversely 



(I) 



Or if 



a h c y y a b c „y y 



= = = = == then m3= = =-=m^= = = 



ooooo 000 00 



mo 

 o 



o 

 o 



Also Ap^^Ar~o, 



(2) 



6'4. I have only sufficient space to indicate the procedure by a simple 

 example. Draw any triangle a, b, c in the ^'-circle and a perpendicular p 

 between one of the angles a and the opposite side a. Draw circles round the 



c-circle 



Fig. 4. 



other two sides b and c as diameters: then the perpendicular^ is the radical- 

 axis of these two latter circles. Let the lines be directed any way we please — 

 as, for example, in Figure 4. Let p divide a into two parts q and 5 ; and 

 den6te reversed chords (not necessarily left chords) by accents. Then we 

 have in Figure 4 three right-rotation triangles which are, by the useful 

 formula of 6-04, 



^ b~b\ b~o)' 



y y V 



o 

 y 



c c 



o 

 c 



What are the relations between these six lines ? The angles /3 and y are 

 common to two versors (chord-ratios) each. 



Hence 

 Also 

 And 

 and 



Therefore 

 9 



C C o 



=o = =o = /v;o 



V ^ c 



p==b 



K\-'b 



bKi 



,1 — ■ 



'. K^o = y sin y sin ^. 

 y 



b cos [\~~y)~y sin j3 sin y. 



=c = Kr 



b' 



' P 

 P 



S==tO 

 



^ c p 

 a = q + s = tO + ro =^ Kc 



c = cK^ I = c cos ( — i3). 



=,0 (by 5-31) = Kb = Kb ^b = b cos ( - y). 



c + Kb b = etc. 



