The quantities x and y will then be the components of ob parallel to oX, o Y.

Since turning a step through two right angles is reversing it, 2=-1; thus i is a value of √(−1).

The operation x + yi is called a complex number.

The ratio ob: oa, which is +√√(x2 + y2), is called the modulus of the complex number x+yi.

=

If a point moves in a plane so that p qo, where q is a constant complex number, it will describe a curve which is called the logarithmic spiral. The velocity of the point p makes a constant angle with op and is proportional to it in magnitude. Let q=x+yi, then x.op is the component of the velocity in the direction op. If r denotes the length op, we shall have rar, and therefore r = ae**, where a is the value of r at the

beginning of the time. Thus the magnitude of op increases at the log. rate x. The component of velocity perpendicular to op is yi.op; it is equal in magnitude to op multiplied by its angular velocity, or (if @ is the angle Xop) it is op. 0. Hence y or the angular velocity is constant. Thus the motion of p is such that op increases at the log. rate x while it turns round with the angular velocity y. Since 0=yt, while r=ae*, it follows that raeke, where k = xy-cot opt.

=

The position vector p of this point may be said to increase at the logarithmic rate q, because p =qp. Hence we may write paest, where a is the value of p when t=0.

The meaning of est, when q is a complex number, is the result of making the unit step oa grow for t seconds at a rate which is got from the step at each instant by multiplying it by the complex number q. In other words, we must make a point p start from a and move always so that its velocity is q times its position-vector; that is, its velocity must be got from the position-vector by turning it through a certain angle and altering it in a certain ratio.

We may now prove that, just as e* is equal to the sum

SERIES OF COMPLEX NUMBERS.

87

of the series f(x), so e is equal to the sum of the series f(qt). To make our former proof available, we have only to premise some observations on complex numbers and on series formed of them.

A complex number q alters the length of a step oa in a certain ratio (the modulus) and turns it round through a certain angle, so converting it into ob. Suppose that another complex number q, turns ob into oc, by altering its length in some other ratio and turning it through some other angle. Then the product q,q is that complex number which turns oa into oc; it therefore multiplies oa by the product of the two ratios, and turns it through the sum of the two angles. Hence qqqq,; or the product of two complex numbers is independent of the order of their multiplication; and the modulus of the product is the product of the moduli. The same thing is clearly true for any number of factors.

=

Instead of operating on a step with a complex number, we may operate on any plane figure whatever. The effect will be to alter the length of every line in the figure in a certain ratio, and to turn the whole figure round a certain angle. Thus the new figure will be similar to the old one. Taking for this figure a triangle, made of two steps and their sum a+B, we learn that q(a+B)=qa+qB. The steps themselves may be represented by complex numbers, namely their ratios to the unit step. Hence also (a+B) q=aq+Bq. Thus complex numbers are multiplied according to the same rules as ordinary numbers.

A series of complex numbers may be divided into two series by separating each term x+yi into its horizontal (or real) part a and its vertical part yi. Neither of these parts can be greater than the modulus of the term; and therefore both parts will converge independently of the order of the terms if a series composed of the moduli converges. To change the series f(qt) into the series of the moduli, we have merely to write mod. qt instead of qt; viz. the series of the moduli is f(mod. qt); because the modulus of q" is the nth power of the modulus of q.

We have before noticed that when the step p grows at the complex log. rate x+yi, its length or modulus r grows

at the log. rate x.

zero.

Hence Ρ is either never zero or always

It may now be proved successively that the series f(gt) is convergent; that if to, t1, t2, t are four quantities in ascending order of magnitude, the mean flux

M = f (gt,) — ƒ (qts)
t1-t2

differs from qf(qt) by a complex number whose horizontal and vertical parts are severally less than the corresponding parts of qf(qt) - af (qt), whose modulus may therefore be made less than any proposed quantity by making t-to small enough; and consequently that the flux of ƒ (qt) is qf(qt). Hence it follows that f(qt)=e, because they both grow at the log. rate q, and are both equal to 1 when

t = 0.

When the velocity of p is always at right angles to op, the logarithmic spiral becomes a circle, and the quantity q is of the form yi. Suppose the motion to commence at a, where oa 1, and the logarithmic rate to be ; that is, the velocity is to be always perpendicular to the radius vector and represented by it in magnitude. Then op=e".

=

m a

Now the velocity of p being unity in a circle of unit radius, the angular velocity of op is unity, and therefore the circular measure of aop is t. But

Therefore

op= = om+mp = cos t + i sin t.

eit =

= cos t + i sin t,

Euler's extremely important formula, from which we get at once the two others,

cost = (e"+e"), i sin t=(e" - e-").

Moreover, ou substituting in these formula the exponential series for e" and e", and remembering that =-1, we find series for cos t and sin t, namely,

=

fi. cd,

The first term is oa; then abi, bci.ab, cdi. bc, de effi.de, and so on. The rapid convergence of the series becomes

manifest, and the point ƒ is already very close to the end of an arc of length equal to the radius.

QUASI-HARMONIC MOTION IN A HYPERBOLA.

It is sometimes convenient to use the functions (e* − e), called the hyperbolic sine of x, hyp. sin x, or hsx, and (e+e*), called the hyperbolic cosine of x, hyp. cos x, or hc x. They have the property hc2x-hs2 x = 1. Thus whenever we find two quantities such that the difference of their squares is constant, it may be worth while to put them equal to equimultiples of the hyperbolic sine and cosine of some quantity: just as when the sum of their squares is constant, we may put them equal to equimultiples of the ordinary sine and cosine of some angle.

The flux of her is hs x and the flux of hs x is a hex, as may be immediately verified.

The motion pa bc (nt + e) + ẞhs (nt + e)_has some curious analogies to elliptic harmonic motion. Let ca = a, cb=6, then cm = ca. hc (nt + e), mp=cb.hs (nt+e), so cm3 mp2 1, or mp2: ma. ma' : cb2: ca2. The curve

that

ca*

cb2

=

=

having this property is called a hyperbola. We see at

once that

then

p = nx hs (nt + e) + nẞ hc (nt + e) = n. cq, say;

cp + cq = (a + B) eo,

and cp-cq = (a-B) e- where =nt + e. Thus pq is parallel to ab, and cn (where n is the middle point of pq) is parallel to ab'. Moreover pn. cn=product of lengths of a+ẞ and a-B=1cx.cy. Hence it appears that the

further away p goes from cy, the nearer it approaches cx, and vice versa. The two lines cx, cy which the curve continually approaches but never actually attains to, are called asymptotes (άσúμπтwтαι, not falling in with the curve). It is clear that the curve is symmetrically situated in the angle formed by the asymptotes, and therefore is symmetrical in regard to the lines bisecting the angles between them, which are called the axes. It consists of two equal and similar branches; though the motion here considered takes place only on one branch.

The acceleration pn'p; thus it is always proportional to the distance from the centre, as in elliptic harmonic motion, but directed away from the centre. The lines cp, cq, are conjugate semidiameters of the hyperbola, as are ca, cb. Each bisects chords parallel to the other, as the equation of motion shews. The locus of q is a hyperbola having the same asymptotes, called the conjugate hyperbola.

The hyperbola is central projection of a circle on a horizontal plane, the centre of projection being above the lowest, but lower than the highest, point of the circle. Let b, a be highest and lowest points of the circle, v the