OPERATIVE ROOTS OF THE CIRCLE-FUNCTION 121 



and so on ; the expansion being like that of a binomial raised to an algebraic 

 power except that powers of O do not appear.^ The series can be further 

 reduced ; so that, for instance, 



K^9 == 56^9 _ 550 _ ^s^K^ + 45^0 + 652 J:« - 350 - K^. (4) 



Operating alternately on zero and on y (or i), we get the values of y sin mO 

 and rcosmd which are the well-known ones. Hence we find the exponential 

 values of sin 6 and cos 6 and the series of i'i3. This can be done more 

 elegantly by using operators throughout, but the demonstration (involving 

 limits) is out of place here. 



3' 6. The Pythagorean Proposition is given by an important algebraic 

 property of the iC-operations ; that is to say 



[02 + {K^fWl = [02 + (r2 - o2)]K^ = [r^]Kf = y^ ; 



hence (K^^)^ + {Kl +^y- = r\ ( i ) 



For, any number by itself operating on any subject merely reproduces 

 itself — because [r^]S — [r^O°]S = r^{S)° = r^, whether S be a number or an 

 operation. The only operation which can be equated to a number is 0° = i. 

 This proposition was really used in equation 3*02. 



The following algebraic results are also frequently required : 



K6.K-9 + K^+<>.K-^-e = Qi-[Kf; (2) 



{K»)^ + {K-^f = 2y2 sin^ 9 + 2 cos 20 . o^ 



{K6f - (X - «)2 = 2 sin 2d.O.K; (3) 



{Ks -cosB. Of = sin^ d . {Kf = (y^ - o^) . sin"^ 6, 



{Kef -2cose.O.KS + 0- = r^' sin^ ; (4) 



the subscript r being understood. 



3*7. The trigonometrical derivation of K^ is obviously as follows. Let 

 a, b, c be the lengths of the sides of any triangle ; let a, fi, y be the opposite 

 angles, and r be the diameter of the circumscribed circle. Then 



a = b cos y + c cos ^ = b cos y + Vc^ — c^ sin^ /3 

 = b cos y + Vr^ sin^ y — b^ sin^ y = K^b ; 

 that is a^ — 2ab cos y + b"^ = c^ = r^ sin^ y. 



Of course the two terms of K^b are respectively the orthogonal projections 

 of b on a, and of Kb on a, the latter being also equal to the orthogonal pro- 

 jection of c on a. 



3*8. What is the relation between K^ and Kl ? Let a = K^b and 

 p = mr. Then 



ma = mK^b = cosy .mb + sin y . Vm^r^ — mW 

 = Kl^^mb = Kpmb ; 



that is Klb - ^,Kl{^b) 



«;=P''>;][^]"- 



y c 

 For example, from 3-7, a = K^b = -i^\yb, where c = y siny. 



^ The reason for this is that [0]« = O when n is an integer. 



