OPERATIVE ROOTS OF THE CIRCLE-FUNCTION 131 



showing that K is invariant between H"^ and H*K Also 



O^ - {Hrf = ^2^0'+ {Krf : (5) 



and, as KrHg = KgHy generally, Kg = KoHq, where Ko = V — i .O. (6) 

 The trigonometrical ratios are compared as follows : 



sin O ■= O-^ cosec O = [i — O] covers O = K—^ cos O = K—^q-^ sec O 



-^ O 



= o-'H-^ cot O = O- W-^O-^ tan O {= j^ tan O). ( 



„, . , . sec o , sec o sec O siw o , x, , O 



Thus if we reqmre we have =■= -^~ =-==-^ = 0~ ivO"" =t7- 



^ cosec O cosec O siw o cosec O ■" 



tow O 

 Similarly H =- =^ ; and O"^ = O-O-^O^; i - 0= 0-^[i + 0]0-'[0- i] O-'; 



and K = 0-m-^0-mO-''. (8) 



Trigonometrical solutions (including inversions) are generally most easily 



dealt with in this way; but I also employ another symbol ^= — 0~^( = — ~), 



which has operative roots resembling those of K ; for, if = w in Quadrantal 

 Measure, then 



sin 6 . T — cos d 



7'1. In applying operative methods to Cartesian Geometry, I denote a 

 curve, not by a function y = <^x, but by the operation (p alone. Thus O is 

 the mid-axis {y = x) ; oo and ooo are the axes of x and y ; a + &o is a straight 

 line ; and Ki is the circle of unit radius described round the origin as centre. 

 To change a unit-circle into an y-circle we attach the alae roand y — 'o. To 

 transfer the circle so that its centre shall be at the point {X, Y) we attach the 

 additional alae Y + O and o — ^ ; so that now 



y = [y + O] [rO] [K] [ro]-'[X + O]-^^, (i) 



and X'^IX + O] [rO] [K]--" [ro]-^ [Y + o]-'y. (2) 



The ellipse with centre at origin, namely mK, can be diminished or enlarged 

 or translated in precisely the same way ; and so can H or any other curve <^. 

 To rotate about the origin through the angle 6, we have, if ^^ is the curve 

 when rotated, 



[cos d . O + sin 6 . (j)g] [cos 6 . O — sind .(j>] = o. (3) 



For ii y = <f)X and y' = (pgX, then by the familiar rules 



y = Kj^x = K^j^y' = K^K+'x'. 

 y = K^x' ^ K^'y = K'^'x : (4) 



where R- = [O^ + (0)2],r = [o^ + {4,Qf]x', and R is the radius-vector of a point 

 on <^ and of the similar point on (^q. To find (^^ we must solve the equation 

 (3). To rotate the axes, put —Q for in the above; and changes of 

 co-ordinates are effected in much the same way. Easy geometrical con- 

 structions exist for illustrating - ^ <^«, 0\//', . ■v/i', etc.; and equations are 

 solved when necessary, by iteration or operative division as shown in my 

 papers 2 and 3. 



7*2. Only the principal nth root of K has been considered here ; but it is 

 clear from 3*03 and from 4'26 that other values exist ; and they seem to be 

 obtainable in just the same way as the wth algebraic roots of K^i — so that, 



for instance, K has the two operative square roots K' and K' (or — K-), and 



