agreed that when Q traverses some specified distance along its -straight path P shall pass through a definite portion of its curved path; whence it will then follow that to every position of Q corresponds a known unique and determinate 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 (J£ 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; 6 increasing meanwhile from o to J A or from o to JA plus a multiple of 2ir. 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 /3; that is, if K = modulus, K = n / I/a2 + a>* . (cos/3+ zsin£) (ii) . The base is the value that OP, or x -+- iy, assumes at the 1nstant when OQ becomes 1, that is, when Q passes through the point J 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= expK (u + iv), or with respect to the base B, as expressed by the identity x + Ty = £"+'v. (iv). Inversely, OQ is the logarith?n of OP, either with respect to the modulus *, as expressed by the identity u + iv=\ogK(x + iy), or with respect to the base B, as expressed by the identity u + iv = B\og (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 circufifertence of the unit circle and the real axis at J. Hence y — o and x = r, when u — v = o, to which correspond the convenient relations X- expKo = B°= 1, and l°g/c.1 = B l°g 1=0. Here also, as in Art. 23, because w= 1 when z=?B, .-. \ogK B — 1 > and exPx 1 = = 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 gome, or a-gonic, according as its modulus does or does not involve the angular element /3. 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, around the origin as a fixed centre, backwards through the angle JOT, so that T, S fall into the positions J, S'. D and E then coincide upon JD, OF becomes perpendicular to JD, — ji remains unchanged in value but merges into <f>, and the new modulus becomes /i / y/ X, -f- co,, or /* / X • cos <f>, its former value with the factor cis ji omitted; while no change in X, /u. and o> need take place. Thus the path of P remains intact, and the new Q moves with its former speed in a straight line passing through S' and through a point on OJat a distance to the left of O equal to OE. Hence the values of z in the two systems are identical, while w, of the original gonic system, in virtue of the back- . ward rotation through the angle JOT, is transferred to the new or agonic system, by being multiplied by cis (— /3), so that the original w and its transformed value, here denoted by w', bear to one another the relation w — w' cis ji. :W 'lA^' The agonic system above described obviously has for its equations of definition (Art. 68) w'—\ogmz, z = buf, in which m — /* / X • cos and b is the value of z for which \ogmz= 1. The formula connecting logarithms in the two systems therefore is log* ^ == cis /3 • logm z. [k = m cis 3- ] Finally, if in this equation z — B, the resulting relation between ji and B is cis(-/3) = log„^, or in the inverse form it is 71. Special Constructions. A further specialization is obtained by making <jb — ji = o. Q then moves upon — a line parallel to OJ, and since tan (<£ — ,3) is now zero, a> ^ is also zero, H remains fixed at A, and P moves upon a f\ straight line passing through O and A. This is a convenient construction for logarithms and exponentials of complex quantities with respect to a real modulus. The base is here also real (Art. 70). In particular, if Q moves upon the real axis itself, A coincides with J, P also moves upon the real axis, and the ^ resulting construction is that described in Art. 23 for logarithms of real quantities. Returning to the general case in which /3 is not zero, we are at liberty, by our original hypothesis concerning the motion of Q, to permit Q to move upon any straight line in the plane, provided we assign to P such a motion as shall consist with the definition that OQ shall be the loga- \ rithm, to modulus K, of OP. (Accordingly, let Q be sup- 3 J posed to move parallel to FN'• tf> — /3 is then zero, o> is likewise zero, and P moves upon a straight line passing through Q and A. | Or again, let Q move parallel to OP; <t> — ji is then ir / 2, X is zero, and P moves in a circumference concentric with the unit circle. Such constructions ^ are possible to every logarithmic system and enable us to simplify tbe graphical representation of the relative motions of P and Q* Expressions of the form logK x, for which no interpretation could be found in terms of real quantities unless x were real and positive (Art. 57), will henceforth be susceptible of definite geometrical representation for all possible values of x. See the examples of Art. 86. *We might, in fact, propose to assign, as the path of Pin the first instance, any straight line passing through the origin, define OP as the exponential of OQ. and then determine the modulus by the appropriate auxiliary construction. 72. Relative Positions of A and C in Fig. 36. We have by definition, c=OC, a=J A -f- a possible multiple of 3ir, The product of the last two of these equations gives m o> = /j. sin (<£ — fi). But o> and //. sin (<£ — /3) are the rates of change of 6 and z,' respectively, and 6 = 0, v'=o are simultaneous values (Art. 68), .'. md = v'; and since 6 = a, v'=c are also simultaneous values (Art. 68), . ma = c. Thus A has always a definite positipn depending upon the modulus and the distance from the origin at which Q crosses the modular normal. / Since a is the length of arc over which J? passes while Q passes from E to C, it is evident that when c / m lies between 2 k w and 2 (k -\- 1) ir, say c I m - 2 kir -\- «2 if) where k is an integer, the part of P's path that corresponds to EC encircles the origin k times before it intersects the circumference of the unit circle, and the point upon the real axis that corresponds to E is its (k + 1)th intersection with the path of/', counting from J to the left. 73. The Exponential Formula. If 6 = arc-ratio of MOP, z=p c'\s6 = Bu + TM. It is required to find p and 6 as functions of u and v. |