vw=v log1z=logyÊ2, in which is any complex quantity, and we may reiterate for gonic systems of logarithms the first proposition of Art. 24: (i). To multiply the modulus of a logaritnm by any quantity has the effect of multiplying the logarithm itself by the same quantity. Corresponding to this equation connecting logarithms in two systems whose moduli are κ and vê, the inverse, or exponential relation is Let C be the base in the system whose modulus is v«; then the following equations co-exist: W= log, z, z=expÂw=. Bw, vw=logy, z= exp1 vw = Cvw, in which are involved, as simultaneous values of w and 2, W = I, w=1/v, when z= = B, (Art. 68 (ii).) These pairs of values, substituted successively in the fourth and second of the previous group of equations, give, as the relations connecting B, C and v, Hence we may reiterate for gonic systems of logarithms the second proposition of Art. 24: (ii). If the modulus be changed from к to vк, the corresponding base is changed from B to B. The third proposition of Art. 24 is a corollary of this second; for if the modulus be changed from < to 1, the base is thereby changed from B to B; that is, (iii). The exponential of any quantity with respect to itself as a modulus is equal to natural base. Finally, if B be substituted for z in the equation log1 = klnz, which is a special case of the formula of proposition (i), the resulting relations between κ and B and, in terms of its base and of natural logarithms, the logarithm to modulus is K log1 = ln z / In B. 76. The Law of Involution. By virtue of proposition (i) of last Article we have in general Otherwise expressed, since changing « into κ/t, or κ/w, changes B into Bt, or Bw, the statement contained in this set of equations is that Btw =(Bt)w = (Bw)t. Obviously or w, or both, may be replaced by their reciprocals in this formula, and the law of involution, more completely stated, is (Bxt)\w=(Bxw)xt. (Cf. Art. 25.) 77. The Law of Metathesis. Let z= by the law of involution zt = (Bw)t = Bwt, and to these there correspond the inverse relations w=log× 2, wt=log; whence the law of metathesis, flog, z=log Z1. Bw; then (Cf. Art. 26.) Also, by changing the modulus (Art. 75) we may write (Cf. Art. 36.) t log w z = w log, z. 78. The Law of Indices. Let w and t be any two complex quantities, w=u+iv, t=r+ is, in which u, v, r and s are real. By the exponential formula (Art. 73), in which m and b correspond to one another as modulus and base respectively in an agonic system of logarithms, and are both real. Hence, by the laws of geometric multiplication and division (Arts. 60, 61), But, by the laws of geometric addition and subtraction (Art. 58), (u±r)+i (v±s) = (u + iv) ±(r+ is) Thus the law of indices obtains for complex quantities. (Cf. Art. 27.) 79. The Addition Theorem. Operating upon both sides of the equation last written with log, we have log (BwBt)=w±t, which, if Bwz and Bt=2', and therefore z=log w and z'= log, t, becomes log (zz')=log ̧2±log ̧2′, K which is the addition theorem for complex quantities. (Cf. Art. 28.) = 80. The Logarithmic Spiral. The locus of P, in Fig. 36 of Art. 68, is obviously a spiral. Its polar equation may be obtained by considering the rates of change of þ and 0, which are respectively λp and wλ tan (Þ — 3). By the definition of a logarithm for real quantities (Art. 23), is here the logarithm of p with respect to the modulus λ tan (p − 3) / λ = tan (ø — 3); or in equivalent terms, 0=tan (4 — 3) · Inƒ, which is the equation sought. w= This locus is called the logarithmic spiral. It is obvious from the definition of the motion of the point that generates this curve (Art. 68), that the spiral encircles the origin an infinite number of times, coming nearer to it with every return, and that it likewise passes around the circumference of the unit circle an infinite number of times, going continually farther away from it. 81. Periodicity of Exponentials. From its character as an operator that turns any line in the plane through the angle whose arc-ratio is 0, it is evident that cis acquires once all its possible values corresponding to real values of , when passes continuously from o to 27 and, k being any integer, repeats the same cycle of values through every interval from 2kπ to 2 (k+1), so that for all integral values of k, cis (0+2)= cis 0. In consequence of this property, cis is said to be periodic, having the period 2π. The exponential Bw has similarly a period. Solving for u and v the two linear eqations, or this result may be inferred by direct inspection of Fig. 36. Hence the equation Bu+wvbu cis v'/m, of Art. 73, may be written m = In this formula let z'o and v' 2km, k being any integer. Then, since cis 2kr=1, Hence Bw has the period 2iкT. In particular e has the period 22π. |