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, - ẞ remains unchanged in value but merges into 4, and the new modulus becomes μ/√λ2 + w2, or μ/cos, its former value with the factor cis ẞ omitted; while no change in λ, μ and w need take place. Thus the path of P remains intact, and the new

moves with its former speed in a straight line passing through S' and through a point on OJ at 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 backward rotation through the angle JOT, is transferred to the new or agonic system, by being multiplied by cis (- ẞ), so that the original w and its transformed value, here denoted by w', bear to one another the relation

w=w' cis ẞ.

The agonic system above described obviously has for its equations of definition (Art. 68)

w'=logm2, z=bw',

in which m= μ/λcos and b is the value of z for which logm 2 = 1. The formula connecting logarithms in the two systems therefore is

log = cis ẞ·logm2.
K

[K=m cis B.]

Finally, if in this equation z=B, the resulting relation between B and B is

cis (-3)=logm B,

or in the inverse form it is

B= expm {cis (-3)}=bcis(8),

71. Special Constructions. A further specialization is obtained by making =ß=0. Q then moves upon a line parallel to OJ, and since tan (Þ — 3) is now zero, w is also zero, R remains fixed at A, and P moves upon a 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 ẞ 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 logarithm, to modulus κ, of OP. Accordingly, let Q be supposed to move parallel to EN'; ☀ — ß is then zero, w is likewise zero, and P moves upon a straight line passing through O and A. Or again, let Q move parallel to OF; ẞ is then π / 2, λ 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 the graphical representation of the relative motions of P and Q.*

Expressions of the form log, 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=JA+ a possible multiple of 2π,
m=μ/λcos ( — ẞ),
- λ tan ( — ẞ).

@=

The product of the last two of these equations gives

But w and μ sin (B) are the rates of change of and v respectively, and @=o, v′=0 are simultaneous values (Art. 68),

.'. m0 = v′;

and since 0: =a, v'= c are also simultaneous values (Art. 68),

Thus A has always a definite position 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 R passes while Q passes from E to C, it is evident that when c/m lies between 2 and 2 (k+1), say

c!m=2kπ + (<2)

where 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 P, counting from / to the left.

73. The Exponential Formula. If Ꮎ of MOP,

=arc-ratio*

It is required to find p and ℗ as functions of u and v.

Since the speed of N' in OT is μ cos (☀ — ß) and that of P in OR is λp, and since by definition m=μ /λ ⋅ 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

But by considering the projections of u, v upon u', v', in Fig. 36, we easily discover that

Or since be1m (Art. 24), this formula may be written

and when m=1 and therefore b e, it assumes the more special form

eu+iv = eu (cos v + i sin v),

an equation due to Euler.*

*Introductio in Analysin Infinitorum, ed. Nov., 1797, Lib. I, p. 104.

74. Demoivre's Theorem. When u=0, Euler's formula becomes

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 = k* 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 log = log cis y 2.

K

In a second transformation, let the motion of P still remain undisturbed, while the speed of Q is changed from μ to nu (na real quantity). By this change the modulus κ cis y becomes nk cis y, the distance of Q from the origin becomes ng instead of q, and wcis y is transformed into nw cis y. Hence, writing n cis y = v, we have

* Demoivre: Miscellanea Analytica (Lond., 1730), p. I.