OPERATIVE ROOTS OF THE CIRCLE-FUNCTION 125 



origin normal to a ; but it is not of the same length unless a = J. By 3-11, 

 E^a = — a, and is a chord in the negative circle equal in length to a but drawn 

 from the origin in the opposite direction ; and K^a is the negative of K^a 

 and equals K~^a, both being in the negative circle. Similarly K~^, K~^ 

 turn a chord through two, or three, negative right angles. 



(5) Of course any chord may be considered to be the radius-vector 

 (capable of negative values) of the double circle of Figure i referred to polar 

 co-ordinates with the pole at the origin and the initial line along the positive 

 tangent t. 



(6) In 3'0 it was shown analytically that K'K* equals either JFf + * or 

 K^-"^, the ambiguity being due to that of the sign of Vr^ — O^ (using K here 

 in a purely algebraic sense and not as subsequently defined). The explana- 

 tion of this is now easy ; for as two counter-chords of the same length may be 



p'=K^a' 



p=Kya 



q'^K^a' 



q=K-ya 



Fig. 2. 



drawn from any point of a circle it follows that operation on both of them 

 by the same operator K"* will have different results. Let a be a right chord 

 and a' its left counter-chord ; let y be positive ; and let p = K^a and 

 q' = Kya'. Then ^ = K> + "o, and q' = Ky+^'-'^o. That is q' = K^-^'^-y^o, 

 and is a left chord and the counter-chord of q where q = K'^~yo ; while the 

 counter-chord of p is p' = K'^-'^-yo. Hence the analytical equations of 3'0, 

 being concerned only with lengths, are, so to speak, forced to give ambiguous 

 results (Figure 2). 



(7) With this explanation we see that K* always turns a subject chord in 

 the same direction — namely counter-clockwise if d be positive. 



5*0. We may now touch upon the operative ratios and operative logarithms 



of chords suggested in i-ii and 1-14. 



If a and b be two fixed numbers, such as the co-ordinates of a point, then 



a 



T { = ({>) denotes any curve ^ passing through that point ; but if y and x be 



y 



variables susceptible of all values, then - denotes a specific curve (f> passing 



through an infinite series of points (see 7'i), 

 If a//b = <^ and c//d = V' then 



a c 

 b d' 



c a 



= <^y\r = ^<it 



only if <J) and ^ be permutable. 



d b 

 This is the case if (P 



(I) 



V* and if ^ = x*' 



