128 SCIENCE PROGRESS 



If we take the same figure in left rotation we must replace all the chords 

 in the above equations by their counter-chords ; so that 



A'B'C'D' . . . = R^»-\ (5) 



Symmetrical formulas in terms of the chords are 



abc ro"l«-2 a' b' c' rol'-" 



= =•>...= = . = = = ...= = . 6 



r r r [_rj r r r \_rj 



The reader can easily deal with re-entrant polygons. Suppose that we 

 already have the right-rotation polygon A BCD = R^ and wish to add an out- 

 side triangle in place of the side D, then this triangle must be (by the following 

 section) D'EF = R^, or D = EF. Hence the enlarged polygon is A BCEF = i?*. 

 If the added triangle is to be re-entrant, we shall have DEF = R^ ; whence 

 ABC = EF, or ABCE'F' = R\ 



6*2. The equation for any triangle in right rotation is 



ABC^R^; (I) 



whence A'^BC, B' = CA, C = AB ; (2) 



R^A=B'C'. R^B = C'A'. R^C = A'B' ; (3) 



or a' = Bc = Cb, b' = Ca = Ac, c' = Ab = Ba, etc., (4) 



give each side in turn, a' and b, b' and c, etc., being co-initial. The various 

 scalar equations for the solution of triangles are supplied by these, by the 

 variations noted in 5-2, and by others which I have no space for. The left- 

 rotation triangle is A'B'C = R*. 



For the right-angled triangle, since one side, say c, must be a diameter, 

 we have 



AB = R, A' = RB, B' = RA. (5) 



a + /3=i, A'B' = RK (6) 



For the isosceles triangle, if the sides a and 6 are equal (they cannot be 

 counter-chords) 



^2 = C = B\ 2a = v' = 2^ ; (7) 



and for the equilateral triangle 



A^=^R\ A'=^A^, 3a = 2. (8) 



Of course in all these figures f is the diameter of the circumscribed circle. 

 The scalars oi a = Ao,b = Bo, . . . give the equations a = r sin a, b = rsin^, 

 . . . which are the properties of all chords of circles, not merely of triangles. 



All the above equations must apply equally to the point-circle and to 

 its infinitesimal polygons (1.12) ; and I denote its operator-chords by 



^0» ^0> ^O' • • • 



All the equations may also be presented in terms of R only, since A = J?*, 

 B = R^, etc. The notation to be employed is suggested by the problem 

 under consideration. It is often easiest to form first the equation between 

 the angles, as in 6-1 1 ; or at other times to commence with chord-ratios. 



When a and b are co-initial the relation r = " = C, where c is the third side 



b 



of the triangle, is especially useful. 



6*3. Hitherto we have dealt only with chords of the same or of equal 

 circles. Those of different circles are easily connected by the relation shown 



in 3'8, between Kl and Kl. Operator-chords of different circles are denoted 

 by Ar, Ap, Br, Bp, etc. ; and Aa, Bb, Cc, . . . are diameters of circles drawn 

 round a, b, c, . . . and are therefore self -reversible. 



It is geometrically obvious that any proposition regarding chords of one 



