m= Since the speed of N' in OT is μ cos (ẞ) and that of P in OR is λp, and since by definition mμ/cos ( — ß), the relation between ON', u' and OP, =p, two real quantities, is that of an exponential to its logarithm, with respect to the modulus m (Art. 23); that is, if b base corresponding to modulus m, u'=logmp, and p = bu' ; and in Art. 72 it was shown that = Hence 0 = v′ | m. Butiv Butiv pass But by considering the projections of u, v upon uʼ, v', in v cos ẞ — u sin ẞ m elm (Art. 24), this formula may be written eu cos ẞ+vsin ẞ)/m cis and when m=1 and therefore be, it assumes the more special form eu+iv = eu (cos vi sin v), an equation due to Euler.* * Introductio in Analysin Infinitorum, ed. Nov., 1797, Lib. I, p. 104. C 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 OJ an angle whose arc-ratio is ẞy, the modulus K, m cis ẞ, by virtue of this change, becomes m cis (B+y)=m cis ß 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·logk2= = In a second transformation, let the motion of P still remain undisturbed, while the speed of Q is changed from u to nu (na real quantity). By this change the modulus κcis y becomes nk cis y, the distance of Q from the origin becomes ng instead of q, and w cis γ is transformed into nw cis y. Hence, writing n cis yv, we have * Demoivre: Miscellanea Analytica (Lond., 1730), p 1. vw= v log ̧2=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: in which are involved, as simultaneous values of w and 2, W = I, when z=B, w=1/v, when z= 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-C', C-B1. 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 κ to 1, the base is thereby changed from B to B; that is, (iii). expк= BK = e; or 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 = ln z, 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 76. The Law of Involution. By virtue of proposition (i) of last Article we have in general Otherwise expressed, since changing κ 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)×t. (Cf. Art. 25.) 77. The Law of Metathesis. Let z= Bw; then by the law of involution 78. The Law of Indices. Let w and be any two complex quantities, wuiv, 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), (vs) cos 3-(ur) sin 3 BwBt=bu±r)cos ß+(v±s)sin ß cis B(u±1)+iv±s). m But, by the laws of geometric addition and subtraction (Art. 58), |