122 SCIENCE PROGRESS 



3' 9. I conclude this section with two important propositions. First 

 it will be seen from the above that these operations are in many ways freely 

 associative and commutative with sines and cosines and with each other. 



Let ^, B, C, . . . denote iC K^. K], . . . for short, so that A' = K'^, 



J5» •= iCf, etc. Then, for example, 



A= AB--^B^cos{a-^).B + sin (a - /3) . KfB. (i) 



Therefore A^ => cos 2 (a - ^) . 5^ + sin 2 (a - /3) . KrB^ 



- {cos^ (a - ^) - sin" (a - /3) } . £2 + 2 cos (a - ^) sin (a - ^) . KrB^ 



- cos" (a - jS) . B2 + 2 cos (a - /3) sin (a - |3) . KfB^ 



+ siyt" (a - ^) . KfB'' ; (2) 



whUe {A )2 = cos* (a - /3) . (B)* + 2 cos (a - /3) sin (a - /3) . B . iiTrS 



+ si«2(«-i3).(is:rB)2; (3) 



so that the operative square (2) is similar inform to the algebraic square (3). 

 For another example 



A^ + B^= [AB--^ + A-''B]AB = 2 cos (a - ^) . AB (by 3.14) ; 

 or A^ + B^ == {cos 2 a + C05 2 /3) . O + {sin 2a ^ sin 2^) .Kr 



= 2 cos (a - i3) . {cos (a + /3) . O + sin (a + /S) . i^r} / 



which is the same. Hence 



A^ - 2 cos {a - ^) . AB + B^ = o : (4) 



while {A)^-2cos{a-l3) .A.B + (B)* = r^ sin {a - /3) (by 3.64). (5) 



From these we have 



[A + B] [C + D] ^ AC + AD + BC + BD, 

 {A+B){C + D)=A.C + A.D + B.C + B.D; 

 [A + Bf ^A^ + 2AB + B\ {A + By- = {A)^ + 2A . B + {Bf ; (6) 



and so on for any exponent. Equation (4) is interesting, because, though 

 real, it is of the unreal algebraic form y^ — 2 cosdyx -{■ x^ = o, or 

 y = C05 6 .X + I sin 6 . x. In fact, when r — o, (4) degenerates to this algebraic 

 statement, and the operative and algebraic equations 4 and 5 become the 

 same. Now, as indicated in 5*2, A, B, C, . . . are the operator-chords of a 

 circle, and when y =-= o that circle is a point-circle — which explains the peri- 

 odicity of " complex " numbers. They are indeed merely special cases of 

 the theorems here outlined. 



Applications to the Theory of Equations are evidently possible, but they 

 cannot be examined here. 



The second proposition is that by an elementary theorem regarding surds, 

 an equation between A, B, C, . . . must break up into two equations between 

 the rational parts and the irrational parts respectively — which is by no means 

 a property of complex numbers only. For example, by 6*2, the equation 

 ABC = KI denotes any triangle in " right sequence," so that 



cos (a + ^ + y) . O + sin (a + iS + y) . Kr^ - O ; (7) 



hence cos (a 4- i3 + y) = — i and sin (a + /3 + y) =- o ; that is, a + i3 +y = 2. 



4.0. For the geometrical interpretation of the jFsT-operations, draw two circles, 

 both of r diameter, touching each other ; and let one circle be called the 

 positive circle and the other the negative circle. Let their point of contact be 

 the origin ; and from it draw positive and negative tangents t and — t, and 

 the two diameters r and — r. Then it is obvious that we can draw from 

 the origin outwards four chords of the same length, one on each side of the 

 diameter of the positive circle {a and a' in Figure i) and one on each side of 

 the diameter of the negative circle (— a and — a'). 



