UNIVERSITY OF CALIFORNIA, DEPARTMENT OF CIVIL ENGINEERINU the algebra of COMPLEX QUANTITIEŜ. (9). (−1+il√3)3 = ( — § — ¿i † √3)3 = 1. (10). Vx+iy = ± ± { √√√ (x2 + y2) +x+ i√√√(x2+y3) (11). [±(1+2) / 1/2]* = [ ± ( 1 − ¿) / √/2]' = — 1. (12). Write down the expression for tensor in each of the above examples. (13). Prove, by the aid of Argand's diagram (Art. 66), that the tensor of the sum of two or more complex quantities cannot be greater than the sum of their tensors; that is, tsr (a + B) ≤tsra + tsr ß. and (14). By definition (Art. 60), tsr (aẞ)= tsra > tsr ß, α amp (aẞ)=ampa± amp ß, and hence no proofs of these properties are called for. Construct the following, applying for the purpose the rules of algebraic and geometric addition and multiplication (Arts. 3, 5, 58, 60): XII. EXPONENTIALS AND LOGARITHMS.* 68. Definitions. In a circle whose radius is unity, OT is assumed to have a fixed direction, its angle with the real axis=JOT, (Fig. 36), OR is supposed to turn about O with a constant speed, Q to move with a constant speed along any line, as ES, in the plane, Palong OR with a speed proportional to its distance from O. Let * The theory of logarithms and exponentials, as here formulated, was the subject-matter of a paper by the author, entitled "The Classification of Logarithmic Systems,'' read before the New York Mathematical Society in October, 1891, and subsequently published in the American Journal of Mathematics, Vol. XIV, pp. 187-194. It was further discussed by Professor Haskell and by the author in two notes in the Bulletin of the New York Mathematical Society, Vol. II, pp. 164-170. speed of P in OR at A=λ, arc-ratio of JOR=0, speed of Qin ES speed of R in JRS OP, OQ=p, q, ON, NQ=u, v, μ, arc-ratio of JDS=, w, arc-ratio of JOT=ẞ, OM, MP=x, y, ON', N'Q=u', v′, OC=c, JA=a (a possible multiple of 27), OT=cos ẞ+ i sin ß=cis ß, m cis ß=K, ET will be called the modular line, and OF, drawn. through the origin perpendicular to ET, will be called the modular normal. In all logarithmic systems the relation W |λ = = tan (Þ — ß) is assumed to exist, and this, together with the equation μ / v/ d2 + w2=m, by elimination of w, gives, as a second Let the values of m and ẞ be assigned, and the path and speed of Q determined, by fixing the angle JDS, the position of the point Cand the value of μ. The value of A is then completely determined through the equation m=μ/λcos (B), and the value of o by the previous equation w= - λ tàn (Þ — ß). Thus the curve upon which P moves, when moves upon a known straight line, is given its definite form by the values assigned to m and ẞ, which are therefore the two independent determining factors in any logarithmic system. But it is still unknown whether, when the position of Q is assigned, P is far or near, and in order to completely define the position of P relatively to that of Q let it be agreed that when Q traverses some specified distance along its straight path Pshall pass through a definite portion of its curved path; whence it will then follow that to every position of corresponds a known unique and deter minate position of P. A convenient assumption for this purpose is found to be that while Q is passing from the modular line to the modular normal (E to C), P shall go from a point on the real axis to the circumference of the unit circle, upon a segment of its path, which, if necessary, may wind about the origin one or more times; 0 increasing meanwhile from o to JA or from o to JA plus a multiple of 27. This assumption determines C and A as corresponding points and fixes A as a definite point on the circumference of the unit circle (see Art. 72). Having thus set up a unique or one-to-one correspondence between the positions of P and Q in their respective paths, we define the terms modulus, base, exponential and logarithm as follows: (i). The modulus is the product of the two independent quantities m and cis ẞ; that is, if x = modulus, x=μ / √x2 + w2 • (cos ẞ+ i sin ß) =μ/cos (B) (cos ẞ+ i sin ẞ). (ii). The base is the value that OP, or x + ¿y, assumes at the instant when OQ becomes 1, that is, when Q passes through the point as it moves along some line that intersects the unit circle at J. In general B will stand for base corresponding to modulus K. (iii). OP is the exponential of OQ, either with respect to the modulus K, as expressed by the identity x + iy=expx (u + iv), or with respect to the base B, as expressed by the identity x+iy=Bu+iv (iv). Inversely, OQ is the logarithm of OP, either with respect to the modulus κ, as expressed by the identity uiv=log(x+iy), or with respect to the base B, as expressed by the identity u+iv=Blog (x+iy). 69. Exponential of o, 1, and Logarithm of 1, B. If the path of Q pass through the origin, the points E and C will coincide at O and the path of P will cross both the circumference of the unit circle and the real axis at J. y=0 and x= 1, when u=v=0, to which correspond the convenient relations and exp ̧¤=B°=I, Hence Here also, as in Art. 23, because w= I when z = B, 70. Classification of Systems. The special value zero for the modular angle JOT eliminates the imaginary term from the modulus and introduces the ordinary system of logarithms, with a real modulus. A system is called gonic, or a-gonic, according as its modulus does or does not involve the angular element ß. The geometrical representation of agonic systems is obtained from Fig. 36 by turning the rigidly connected group of lines EN', EQ and OF, together with the specified points upon them, arcund the origin as a fixed centre, |