of H, between the system (J) and the equation (L4); and the final integrals, of the same system of differential equations (A), being now (theoretically) obtained, by eliminating the same constant H between (K4) and (L4). (g). The functions F and V are obliged to satisfy certain Partial Differential Equations in Quaternions, of which those relative to the final vectors a, a', .. are the following, (DF)-2m (DF)2= P, (M1); }Σm ̄1(D ̧V)2+P+H=0; (N1) a and they are subject to certain geometrical conditions, from which can be deduced, in a new way, and as new verifications, the law of motion of the centre of gravity, and the law of description of areas. (h). General approximate expressions (p. 717) for the functions F and V, and for their derivatives Hand t, for the case of a short motion of the system. ARTICLE 419.-On the Relative Motion of a Binary System; and on the Law of the Circular Hodograph, (a). The vector of one body from the other being a, and the distance being r (= Ta), while the sum of the masses is M, the differential equation of the relative motion is, with the law of the inverse square, Da Ma-1; (04) D being here used as a characteristic of derivation, with respect to the time t. (b). As a first integral, which holds good also for any other law of central force, we have which includes the two usual laws, of the constant plane (+ ẞ), and с TB of the constant areal velocity (= = T3). (c). Writing = Da = vector of relative velocity, and conceiving this new vector to be drawn from that one of the two bodies which is here selected for the origin o, the locus of the extremities of the vector 7 is (by earlier definitions) the Hodograph of the Relative Motion; and this hodograph is proved to be, for the Law of the Inverse Square, a Circle. (d). In fact, it is shown (p. 720), that for any law of central force, the radius of curvature of the hodograph is equal to the force, multiplied into the square of the distance, and divided by the doubled areal velocity; or by the constant parallelogram c, under the vectors (a and 7) of position and velocity, or of the orbit and the hodograph. (e). It follows then, conversely, that the law of the inverse square is the only law which renders the hodograph generally a circle; so that the law of nature may be characterized, as the Law of the Circular Hodograph: from which latter law, however, it is easy to deduce the form of the Orbit, as a conic section with a focus at o. Pages. 717-733 (g). The orbital excentricity e is also the hodographic excentri- city, in the sense that eh is the distance of the centre н of the hodo- graph, from the point o which is here treated as the centre of force. (4). The orbit is an ellipse, when the point o is interior to the bodographic circle (e<1); it is a parabola, when o is on the circum- ference of that circle (e=1); and it is an hyperbola, when o is an ex- the constant a will have its usual signification, relatively to the (1). The quantity Mr being here called the Potential, and de- noted by P, geometrical constructions for this quantity P are assigned, with the help of the hodograph (p. 723); and for the harmonic mean, 21(+), between the two potentials, P and P′, which answer to the extremities T, T′ of any proposed chord of that circle: all which constructions are illustrated by a new diagram (Fig. 86). (j). If u be the pole of the chord TT'; M, M' the points in which the line ou cuts the circle; L the middle point, and N the pole, of the new chord MM', one secant from which last pole is thus the line NTT'; the intersection of this secant with the chord Mм', or the harmonic conjugate of the point u, with respect to the same chord; and NT,T,' any near secant from N, while u, (on the line ou) is the pole of the sear chord TT: then the two small arcs, TT and TT,', of the hodo- graph, intercepted between these two secants, are proved to be ulti- mately proportional to the two potentials, P and P'; or to the two ordinates TV, TV', namely the perpendiculars let fall from T and T', on may here be called the hodographic axis LN. Also, the harmonic mean between these two ordinates is obviously (by the construction) (k). In general, for any motion of a point (absolute or relative, in one plane or in space, for example, in the motion of the centre of the moon about that of the earth, under the perturbations produced by the attractions of the sun and planets), with a for the variable vector (418) of position of the point, the time dt which corresponds to any vector- element dDa of the hodograph, or what may be called the time of ho- dographically describing that element, is the quotient obtained by dividing the same element of the hodograph, by the vector of accelera- sion D'a in the orbit; because we may write generally (p. 724), the times dt, de, of hodographically describing the small circular arcs TT and TT,' of the hodograph, being found by multiplying the lengths (j) of those two arcs by the mass, and dividing each product by the square of the potential corresponding, are therefore inversely as those two potentials, P, P', or directly as the distances, r, r', in the (m). If we suppose that the mass, M, and the five points o, L, M, (n). And hence may be deduced (p. 726), by supposing one secant to become a tangent, this Theorem of Hodographie Isochronism, which was communicated without demonstration, several years ago, to the Royal Irish Academy, and has since been treated as a subject of investigation by several able writers: If two circular hodographs, having a common chord, which passes through, or tends towards, a common centre of force, be cut perpendicu- larly by a third circle, the times of hodographically describing the inter- (o). This common time can easily be expressed (p. 726), under the 2g being the length of the fixed chord MM'; e' the quotient LO: LM, which reduces itself to -1 when o is at m', that is for the case of a pa- rabolic orbit; e' lying between + 1 for an ellipse, and outside those limits for an hyperbola, but being, in all these cases, constant; while w is a certain auxiliary angle, of which the sine UT: UL (p. 727), or = 8 (r+r') ́1, if s denote the length PP' of the chord of the orbit, cor- responding to the chord TT' of the hodograph; and w varies from 0 to π, when the whole periodic time 2πn-1 for a closed orbit is to be computed: with the verification, that the integral (V1) gives, in this last case, See the Proceedings of the 16th of March, 1847. It is understood that the common centre o of force is occupied by a common mass, M. (p). By examining the general composition of the definite inte- gral (V4), or by more purely geometrical considerations, which are illustrated by Fig. 87, it is found that, with the law of the inverse square, the time t of describing an arc PP' of the orbit (closed or un- and therefore simply a function of the chord (s, or PP') of the orbit, : (4). The same important theorem may be otherwise deduced, (r). Writing now (comp. p. xlvii) the following expression for the 2T=-MDa2 = 2 (P+ H) = M (2r ̄1 — a ̄1) ; F=f*(P+T)dt, (Z.), and V=['2T&t=F+tH, (As) which have thus (comp. (E) and (F1)) the same forms as before, but with different (although analogous) significations, and may still be called the Principal and Characteristic Functions of the motion; and denoting by a, a' (instead of ao, a) the initial and final vectors of po- sition, or of the orbit, while r, r' are the two distances, and 7, 7' the two corresponding vectors of velocity, or of the hodograph: it is found that when M is given, F may be treated as a function of a, a', t, or of r, r', s, t, and V as a function of a, a', a, or ofr, r′, 8, and H; and that their partial derivatives, in the first view of these two functions, (D1) F= - H, (D); and DяV=· DaV= t; while, in the second view of the same functions, they satisfy the two along with two other equations of the same kind, but of the second (s). The equations (F。) (G5) express, that the two distances, r three quantities, r+r', s, and t; while V, and therefore also by (Es), is found in like manner to be a function of the three scalars, r+r', s, and a: which last result respecting the time agrees with (p), and furnishes a new proof of Lambert's Theorem. (t). The three partial differential equations (r) in V conduct, by merely algebraical combinations, to expressions for the three partial derivatives, D, V, D‚' V (= D ̧Ð ́), and D.; and thus, with the help of (E5), to two new definite integrals* (p. 731), which express respectively the Action and the Time, in the relative motion of a binary system here considered, namely, the two following: whereof the latter is not to be extended, without modification, beyond the limits within which the radical is finite. ARTICLE 420.-On the determination of the Distance of a Comet, or new Planet, from the Earth, . 2 (a). The masses of earth and comet being neglected, and the mass of the sun being denoted by M, let r and w denote the distances of earth and comet from sun, and ≈ their distance from each other, while a is the heliocentric vector of the earth (Ta=r), known by the theory of the sun, and p is the unit-vector, determined by observation, which is directed from the earth to the comet. Then it is easily proved by quaternions, that we have the equation (p. 734), Pages. 733, 734 eliminating w between these two formulæ, clearing of fractions, and dividing by z, we are therefore conducted in this way to an algebraical equation of the seventh degree, whereof one root is the sought distance, z. (b). The final equation, thus obtained, differs only by its notation, and by the facility of its deduction, from that assigned for the same purpose in the Mécanique Céleste; and the rule of Laplace there given, for determining, by inspection of a celestial globe, which of the two References are given to the First Essay, &c., by the present writer (comp. the Note to p. xlvii.), in which were assigned integrals, substantially equivalent to (Hs) and (Is), but deduced by a quite different analysis. It has recently been remarked to him, by his friend Professor Tait of Edinburgh, that while the area described, with Newton's Law, about the full focus of an orbit, has long been known to be proportional to the time corresponding, so the area about the empty focus represents (or is proportional to) the action. |