74. Demoivre's Theorem. When « = o, Euler's formula becomes e'v = cis v, and by involution einv — (cis vyt — c;s nv> or (cos v -f- z'sin v)n — cos wz> -f- isinnv for all real values of n. This equation is known as Demoivre's theorem.* O 75. Relations between Base and Modulus. Let the lines EN\ EQ and OF be regarded for the moment as rigidly connected with one another and be turned conjointly in the plane about the fixed point O through an arbitrary angle, whose arc-ratio may here be denoted by y. ON', in the new position thus given it, then forms with 0/ an angle whose arc-ratio is /3 -f- y, the modulus K, — m cis /3, by virtue of this change, becomes m cis (/3 + y) = m cis /3 • cis y = # • cis y, and since OQ, in common with the other lines with which it is connected, is turned about O through the angle of arcratio y, w is hereby transformed into w cis y; while the locus of P is in no way disturbed by any of these changes. Hence w cis y = cis y • logK z = logK cisyz. In a second transformation, let the motion of P still remain undisturbed, while the speed of Q is changed from /j. to n/j. (n = a real quantity). By this change the modulus K cis y becomes nK cis y, the distance of Q from the origin becomes nq instead of q, and w cis y is transformed into nw cis y. Hence, writing n cis y = v, we have vw—v \ogK2 — \ogVKz, in which v 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 K and Vk, the inverse, or exponential relation is expy,C vw — expK w = expK^ rajLet C be the base in the system whose modulus is Vk; then the following equations co-exist: w — logK z, z = expK w — Bw, vw = \o%VK z, z = expVK vw — Cvw, in^rhich are involved, as simultaneous values of w and z, w = 1, when z — B, w—i/v, when .2= C (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, B=CV, C=B"V. Hence we may reiterate for gonic systems of logarithms the second proposition of Art. 24: (ii) . If the modulus be changed from K to Vk, the corresponding base is changedfrom B to Bllv. The third proposition of Art. 24 is a corollary of this second; for if the modulus be changed from K to 1, the base is thereby changed from B to BK; that is, expKK = .Z?K = *?; or (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 logKz = *lnz, which is a special case of the formula of proposition (i), the resulting relations between K and B Eire K=1/lnif, B=e"K, and, in terms of its base and of natural logarithms, the logarithm to modulus * is logK z = In z / In B. 76. The Law of Involution. By virtue of proposition (i) of last Article we have in general explK w = expK w /1, and expK ltw = expK tw. Hence expK tw = expKIt w = expK lat= expKItw 1. Otherwise expressed, since changing K into K /1, or K / w, changes B into B', or Bw, the statement contained in this set of equations is that Btw={Bt)a'—{B,a'y. Obviously t or w, or both, may be replaced by their reciprocals in this formula, and the law of involution, more completely stated, is {B^^^^B^)^. (Cf. Art. 25.) 77. The Law of Metathesis. Let z — Bw; then by the law of involution , and to these there correspond the inverse relations w = \ogKz, wt=\ogKzl; whence the law of metathesis, t\ogKz = \ogKzt. (Cf. Art. 26.) Also, by changing the modulus (Art. 75) we may write t\ogwz= w\ogtz. (Cf. Art. 36.) 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 J are real. By the exponential formula (Art. 73), „ , » , , » . v cos 3 — u sin 3 m Bt = ircosfi+s^fi dsScos3-rsin3 t m 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), r, w r,, J- ^ J- x • » . (v±s')cos3 — (u±r)sm3 BWY/B1 = ft (ii±r)cos jl + (.v±s)sm p CIS i —' ' i L L 'm But, by the laws of geometric addition and subtraction (Art. 58), |