OPERATIVE ROOTS OF THE CIRCLE-FUNCTION 127 



and as the chord-angle of y is unity, we have Ij^R => i . The operative ratio 

 of the zero-chord to itself must be the same as that of any quantity or 



operation to itself- 



a 



Hence also 



o 



O ; that is, the unit of 

 Its chord - 



required 



or 



-that is - = O 

 a 



operation O is the operator-chord corresponding to the zero-chord 



angle is zero. 



Just as we denote two counter-chords by a and a' so we can denote two 



counter-operator-chords by A and A '. 



(These ideas can be extended beyond the present subject.) 

 5*3. The following formulae can be immediately verified and are frequently 

 a' r a' h' ah a' h 



7^ a' T^a' P'^a'' F^a' 



AA' r= BB' = RR' = R^ -^ - O, or A' = -A-'-. Rr ~ - o. 

 AB = BA, Ab = Ba. li A = BC, a = Be = Cb. 

 Ao = r sin a = Oa = a ; Ar = r cos a = Ra = + 's/r- — a-. 



A = R'^ = cosa.O + sina.R; Or . A = Ar . O + Ao . R 



Or . AB = Ar . OB + Ao . RB = Br . oA + Bo . RA =etc. 

 r . ABo = Ar . Bo + Ao . Br = Ar . Ob + Ao . Rb = etc. 

 r.ABr = Ar . Br - Ao . Bo = Ar . Rb - Ao.Ob = etc. 



A . B = Ao . Bo + AB . O = Ar . Br - RAB . R. 

 A . BC - B . AC = Ao . BCo - Bo . ACo =- Ar . BCr - 

 Br.ACr = Co.AB-i-o. 

 [A^ + J52] [C2 + Z)2] = A^a + ^2i)2 + B2C2 + BW^ - 

 2 cos (a - iS) COS (y - 3) . A BCD. 



(See also 3-2, 3-4, 3-5, 3-6, and 3-9.) 



6*0. The geometrical application of the i^-operations naturally com- 

 mences with the subject of rectilinear figures 

 in circles — a definition which includes all 

 triangles. I have space only for a few notes. 



However any chords may be placed in 

 any circle the identity of 4-14 always holds 

 good, namely that 



a = K^-^b 



(I) 

 (2) 

 ' (3) 

 (4) 

 (5) 

 (6) 

 (7) 

 (8) 



(9) 

 (10) 



or 



Ba = Ab. 



Any series of sequent chords (4-0) in a 

 circle may be in right, or positive, rotation, as 

 in Figure 3 ; or in left, or negative, rotation ; 

 or in changed rotations. This does not mean 

 that all chords in right rotation are right 

 chords ; or in left rotation are left chords. 



Fig. 3. 



6*1. If the n chords a. b, c, d, . . . form 

 a convex closed figure in right rotation 

 (Figure 3), then, by drawing chords to all the angles of the figure from any 

 point on the circumference of the containing circle, we see that the sum 

 of the arc-angles (that is, of the chord-angles) of the chords of the figure is 

 equal to two right angles ; so that 



therefore 



and 



Hence A' 



BCD 



