at the log. rate x. zero. Hence p is either never zero or always It may now be proved successively that the series f(qt) is convergent; that if to, t,, t, t are four quantities in ascending order of magnitude, the mean flux = differs from qf(qt) by a complex number whose horizontal and vertical parts are severally less than the corresponding parts of qf(qt) — qf (qt), whose modulus may therefore be made less than any proposed quantity by making t-t small enough; and consequently that the flux of ƒ (qt) is gf(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 7 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 eit. = 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. e* = cos t + i sin t, Euler's extremely important formula, from which we get at once the two others, cos t = (e"+e"), i sin t = 1 (e" - e ̄it). Moreover, ou substituting in these formulæ the exponential series for e" and e", and remembering that 1, we find series for cos t and sin t, namely, may be graphically verified by construction of the several terms of the series 1 + i + žï2+ ¿ï3 +..... The first term is oa; then abi, bci. ab, cd = fi. bc, effi.de, and so on. de = fi. cd, 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. 1. It is sometimes convenient to use the functions (e* — e ̃2), 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 hc2 x-hs2 x = 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 x hcx, as may be immediately verified. The motion p = abc (nt + e) + Bhs (nt + e) has some curious analogies to elliptic harmonic motion. Let ca = a, cb=B, then cm = ca. hc (nt + e), mp=cb. hs (nt +e€), so cm2 mp2 1, or mp2: ma. ma' = cb2: ca2. The curve that ca2 cb2 = having this property is called a hyperbola. We see at once that then = nx hs (nt + €) + nẞ hc (nt + €) = n. cq, say; = cp + cq = (a + B) eo, and cpcq (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 CENTRAL PROJECTION OF CIRCLE. 91 centre of projection, am the projection of ab and рт of qn which is perpendicular to ab. We find a also and an: am= nf: vf; gb a'm=fb: af; multiply these three together, then But therefore an. nb: am. a'm=nƒa . fb : vf2.af. the property noticed above. Making a change in the figure, the same process shews that the ellipse is central projection of a circle which is wholly below the centre of projection. These three central projections of the circle, ellipse, parabola, and hyperbola, are called conic sections; being plane sections of the cone formed by joining all the points of a circle to a point v. CHAPTER III. CENTRAL ORBITS. THE THEOREM OF MOMENTS. THE moment of the finite straight line pt about the point o is twice the area of the triangle opt. Its magnitude is the product of the length pt and the perpendicular on it from o. m Every plane area is to be regarded as a directed quantity. It is represented by a vector drawn perpendicular to its plane, containing as many linear centimeters as there are square centimeters in the area. The vector must be drawn towards that side of the plane from which the area appears to be gone round counter-clockwise. Thus om is the vector representing twice the area opt, p being the near end of pt and m on the upper side of the plane opt. The sum of the moments of two adjacent sides of a parallelogram about any point is equal to the moment of the diagonal through their point of intersection. That is, triangle oad = oac+oab; each triangle being regarded as a vector, in the general case when o is out of the plane abcd. Taking first the special case of o in the plane, we observe that oad ocd + cad+oac; but ocd+cadoab, because the = height of oab is the sum of the heights of ocd and cad, while all three stand on the same base ab or cd. Therefore |