... OB' = OA' +A'B'; mcise × (a +ẞ) =mcis Xa+m · cis @ × ß. This demonstration is in no way disturbed by the introduction of negative and reciprocal signs. The last equation above written is, in fact, the first equation of page 37, and the subsequent equations of Arts. 20, 21 and their proofs remain intact when for the real quantities a, b, c, d, etc., complex quantities are substituted. Hence, writing m⚫ cis 0=y, (±a±ß) ×(±√) =+ (± a) × (± y) + (± ß) × (±y). Here, as in Arts. 20, 21, the sign X is distributive over two or more terms that follow it, but not so the sign /. 66. Argand's Diagram. It is obvious from its definition as here given (Art. 57) that to every complex quantity there corresponds in the plane a unique geometrical figure which completely characterizes it. This figure is known as Argand's diagram,* and consists of the χ A P -X real axis OX with reference to which the arc-ratio is estimated, the imaginary axis OY perpendicular to OX, the directed line OP that represents the complex quantity and the perpendicular PA from P to OX. The axes OX and OY are supposed to be fixed in position and direction for all quantities. Any point P in the plane then determines one and only one complex quantity and one set of line-segments OA, AP, OP, different from every other set. Fig.35. * First constructed for this purpose by Argand: Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques; Paris, 1806. Translated by Prof. A. S. Hardy, New York, 1881. If OAx and AP=y, the complex quantity appears in the form xiy, and if OP=a and arc-ratio of AOP=0, the relations xa cos 0, yasin 0, are directly evident from the figure and we have x+iya (cos + isin 0), tsr (x+iy)=+√x2 + y2, vsr (x+y)= cos i sin 0, amp (x+iy)=arc-ratio whose tangent is y/x. In analysis the complex quantity most frequently presents itself either explicitly in the form xiy, or implicitly in some operation out of which this form issues. 67. Agenda. Multiplication, Division and Construction of Complex Quantities. Prove the following: (1). (a+ib) × (x+iy) = ax― by +i (ay+bx). ax+by+i (bx — ay). x2 + y2 (2). (a+ib)/(x+iy)= (3). (a+ib) + (a — ib)* = 2 (a1 + b1) — 12 a2b2. 2 (4). (a+b)2 - (a+16) ib a+ib = 3. 4iab x2 — y2 — 2ixy. (x + iy)2 x + iy x3 (7). (x + y / i)2 (x2 + y2)2 x3-- 3xy2+i (3x2y — y3) ̧ (8). (−1+✯ √3)2 — — § — i§ √3. ± } { √√√ (x2 + y2)+x+i √√√(x2+y3) − x }· (11). [±(1+i) / 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) ≤ts and tsratsr ß. (14). By definition (Art. 60), tsr (aẞ) tsratsr 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): C 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 mod. live Fig. 36. 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. XI187-194. 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. p. OC=c, JA=a (a possible multiple of 27), μ / √ √2 + w2 =m, OT=cos i sin ẞcis ß, m cis BK, 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 is assumed to exist, and this, together with the equation μ / √/λ2 + w2 =m, by elimination of w, gives, as a second expression for m, Let the values of m and ẞ be assigned, and the path and speed of 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 /λ⋅ cos (☀ — ß), and the value of w by the previous equation w= A tan ( — 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 aigned, P is far or near, and in order to completely define the position of P relatively to that of Q let it be |
