126 SCIENCE PROGRESS 



say, since x**x"* = X^X^- I* ^^^^ ^-^so be the case with all the operative 

 ratios dealt with here, which will always represent some operative power 

 of Kf. 



For all numbers we assume that 



aba b ran-i 



-. = -and- = L=J ; (2) 



but for all chords we also have by 4- 15 



l = K^-^- 5 = ii7-*; (3) 



a c c a a c ajjh alld aces 



therefore Tn = j? = "5T = T/T" ^ ~tTr = =" " T ji ^t^-, (4) 



b a a b a b a//c b//c e e b d ^^' 



where e is any fifth chord. But I shall not here use such expressions as 

 [a] [c] because these require considerable discussion. 



5'1. We may also use operative logarithms as suggested in 1*14, so 

 that 



ace ace 



^""bdj' ' ' " ^& "*■ ^^2 + ^A'J+ • • • 



-a-^ + y-S + e-f+... (l) 



5*2. Since we can always interpolate any third chord between the two 

 chords of an operative ratio, that third chord may be the positive zero-chord 

 if we wish ; so that 



(2) 



(3) 



(4) 



The meaning of this is that we can reduce ratios between any two chords 

 to ratios between each separate chord and the zero-chord ; and that the 

 operative logarithms of these latter ratios are always the corresponding 

 chord-angles. I therefore denote the ratio of any chord to the zero-chord 

 by the capital of the letter which denotes the former chord, so that for 

 instance 



and ^ii , ^AB-'CD-'EF-^ ... (5) 



Hence, while a, b, c ... are directed lengths measured by numbers. A, B, C, 

 . . . are the corresponding K-operations, possessing each a geometrical 

 relation with its respective chord ; and we may usefully call them operator - 

 chords. 



Of course juxtaposition and indices affect ^, B, C, . . . as they aSect 

 all operations — i.e. operatively and not algebraically. 



Two important operator-chords are those of the diameter r and of the 



r 

 zero-chord itself. The former may be denoted by R — so that R^ -^^ Kf] 



