OPERATIVE ROOTS OF THE CIRCLE-FUNCTION 117 



g 



from an algebraic one. Thus K^ is a versor which converts b into a — that is, 

 which turns b through the angle B and also changes its length to equal that 

 of a, another chord of the circle of r diameter. If r = o, then K , that is, 



{cosB + isin6)o, (2) 



is a " complex " versor which affects the infinitesimal chord of a circle of 

 zero-diameter (point-circle) in the same manner. 



We may infer that most of the important theorems which have been 

 based on De Moivre's can be similarly generalised in terms of K. For 



example, we can resolve K^ ±0 into operative factors and thus solve the 



equation K^ ± O = ; and in place of the usual trigonometrical exponential 

 values we obtain 



K^ = e°0 + 6K+\ G'K^ + \ e^K^ + . . .. (3) 



the subscript r being understood. The operative logarithm is another 

 important entity because it expresses the angle of rotation ; and it may be 

 denoted by the abbreviation opl. Thus as K^ = aljb, we write 



^^oplj^p oT=lj^.^ for short. (4) 



Lastly, the similarity of K^ to a quaternion is obvious, and we may surmise 

 that it will be useful in most branches of geometry. 



2*0. The Quadrantal Measure for angles is always very convenient — not 

 only here, but for general use. We denote an angle merely by the positive 

 or negative integer or fraction which indicates its ratio to a right angle. 

 Thus the angles, it, ^, 60°, 45°, 30° . . . are expressed respectively by 2, 

 I, f , L i . . . ; and cos 1 = 0, sin 1 = 1, cos {1 - 6) = sin 6, etc. Using 

 Circular Measure, we should write 



Kf = cos m liT . O + sin m \tt . K^: (i) 



but the Quadrantal Measure enables us to omit the Jtt. 



2*1. The Explicit Operative Notation employed here is easily explained as 

 follows. Let y be any function of x whatever — suppose for example that 



y =-(j)X = a + bx + cx^ -i- . . . + qx^. (i) 



Now, it is most necessary in operative analysis for us to be able to denote 

 any operation (jf) explicitly and in detail without involving x or any other 

 argument, and in such a manner that we may indicate its inversion, its iteration, 

 and its algebraic and operative relations with numbers or with other opera- 

 tions without circumlocution. How can we do so ? There is certainly no 

 method in general employment now ; but there is one, and only one, method 

 available — we can substitute <^° for x in the expression on the right. Then 



= a + 6(^° -F c(0°)2 + . . .+?(</) °)«. (2) 



If both sides of this equation be appUed to the argument x, the original 

 equation is restored. But this notation has never been used (so far as I can 

 find) in consequence of one of the most singular (but unnoticed) fallacies • 

 in mathematics, that (f}° = i. If we substitute this value in the above 

 equation we obtain 



^ = a + b + c + ...+q; (3) 



which is both meaningless and absurd ; so that the algebraic machinery 

 breaks down — clear evidence that there has been an error somewhere. Six- 



1 It is true that (f)°x = x, but not that 4>°x ^i x x. 



