120 ^ SCIENCE PROGRESS 



By supposing that r = i and by writing K for Ki, we have 



K^ sin 6 = sin {y + 6) = K sin y, 



K^ cos 6 = cos {y + 6) = K cos y ; (6) 



so that Ksin6 = sin {i + 6) == cos 6, 



Kcos6 = cos (i + ^) = — sin 6, 

 and cos 6 = K sin 6 and sin 6 = K-'^ cos 6. (7) 



Thus if we had deduced the properties of K^ independently of trigonometry, 

 we could easily have proved the formulae for the trigonometrical ratios of 

 sums of angles by the method used in 3'o. 



3*3. The reader must be warned to be careful regarding the signs. 

 Obviously "sf \ — sivi^Q = ±.cos6 ; but this ambiguity does not exist when 

 we use K. For K^ sin 6 always equals sin {y -{- 6), the sign of which depends 

 upon the magnitude of y + ^ ; so that always 



K sin6 = sin (i + 0) ; K sin (- B) = sin (i - 0) ; 

 K cose-= cos (i + 6) ; K cos (- 6) = cos (i - 6). ■ 



As cos (- 6) = cos 6 and sin (2 - ^) = sinB, we may think erroneously that 

 Kcos[ — 6) = K cos 6 and K sin {2 - 6) = K sin 6 ; but 



KcosO = — sin 6, K cos { — 6) = sin 6; (i) 



and while Ksin 6 = cosO, Ksin (2 - ^) = /C^ + 2 - Aq = K^K^ - ^o = - cosd. 



Similarly K ( - sin 6) = KK^ sin d = K^K sin 6 = - cos e. (2) 



Before putting such trigonometrical expressions as cos B and sin B under 

 if-operators we must know the sign of d and distinguish between the sine of 

 an angle and of its supplement. In this respect the i^-operators are more 

 exact than the trigonometrical ratios. See also 4*2 1. 



3*4. The expression for K^ consists (i) of the two operations o and K, 

 and (2) of their numerical coefficients cos 6 and sin 6. If we operate on it with 

 K^ we can suppose that this affects either O and K, or their coefficients ; so 



that KlKl =^cosB.Kl-\- sin B.K\ + '^, (i) 



or = cos (y + ^) . O + sin {y + 6) . K^ ; (2) 



or rKlKl = K^r . K^ + K^o . KI + '' = Kl + ^'r .O + K^^'^o . K^, etc. (3) 



3*5. As an example of all this consider the expression for multiple 

 angles. By 3' 14 (omitting r for convenience) 



[K« + K -9]K« = [2 cos ^ . O ]X« = 2 cos ^ . Ko. 

 But [K« + K -6]K0 = O + isT^e ; 



therefore K^» = 2cosB .K& - O. (i) 



Now put s = 2 cos 6 for short, and suppose n to be any integer. 

 Then K^"* = [sK« - O]" and K^''& + » = [sK^ - 0-]»K^ (2) 



are the general expressions. To find the expansions of these we must iterate 

 the expression in square brackets, and for this we shall require to know the 



value of [K^] [sK^ — O] or, as we can write it without ambiguity, 



K^{sK^-0). As sK^-0 = K^^. and we have K^K^^ = K^^K^, the two 

 operations are permutable. Hence 



K^isK^ - O) = s7<29 _ K^, 

 and K*« = [sK^ - o]^ = s^/C^a _ 2sK^ + O, 



K69 ^ ^sK» _ o]3 = s'K'iS _ 3s2J^29 + 352^9 _ o, (3) 



