CHAP. III.] LAW OF THE CIRCULAR HODOGRAPH.

719

orbit, with yɛ (or + εy) for the vector of its centre, and with Ty= MTB-1 for its radius, which radius we shall also denote by h.

(6.) The Law of the Circular* Hodograph is therefore a mathematical consequence of the Law of the Inverse Square; and conversely it will soon be proved, that no other law of central force would allow generally the hodograph to be a circle.

(7.) For the law of nature, the Radius (h) of the Hodograph is equal, by (1.) and (5.), to the quotient of the attracting mass (M), divided by the double areal velocity (TB or c) in the orbit; and if we write

this positive scalar e may be called the Excentricity of the hodograph, regarded as a circle excentrically situated, with respect to the fixed centre of force, o.

(8.) Thus, if e <1, the fixed point o is interior to the hodographic circle; if e = 1, the point o is on the circumference; and if e> 1, the centre o of force is then exterior to the hodograph, being however, in all these cases, situated in its plane. (9.) The equation VII. gives,

operating then on this with S. a, and writing for abridgment,

XII... p =ẞy ̄1 = M-1Tẞ2 = c2 M ̄, and XIII... SUaɛ = cos v,

so that p is a constant and positive scalar, while v is the inclination of a to - ɛ, we find,

the orbit is therefore a plane conic, with the centre of force o for a focus, having e for its excentricity, and p for its semiparameter.

(10.) And we see, by XII., that if this semiparameter p be multiplied by the attracting mass M, the product is the square of the double areal velocity c; 80 that this constant c may be denoted by (Mp)}, which agrees with known results.

(11.) If, on the other hand, we divide the mass (M) by the semiparameter (p), the quotient is by XII. the square of the radius (MTB-1 or h) of the hodograph. (12.) And if we multiply the same semiparameter p by this radius MT3-1 of the hodograph, the product is then, by the same formula XII., the constant TB or c of double areal velocity in the orbit, so that h Mel = cp !.

=

(13.) If we had operated with V. a on VII'., we should have found,

XVI... ẞ= V.a (ε - Ua) y = (Saɛ +r) y ;

which would have conducted to the same equations XIV. XV. as before.

*This law of the circular hodograph was deduced geometrically, in a paper read before the Royal Irish Academy, by the present author, on the 14th of December, 1846; but it was virtually contained in a quaternion formula, equivalent to the recent equation VII., which had formed part of an earlier communication, in July, 1845. (See the Proceedings for those dates; and especially pages 345, 347, and xxxix., xlix., of Vol. III.)

but

and

(14.) If we operate on VII. with S. a, we find this other equation,

if we write

XVII. . . - rDr = SaDa = y Vaε;

XVIII... - y2 = h2 = (by VI. and XII., comp. (11.)),

M
P

hence squaring XVII., and dividing by r2, we obtain the equation,

(15.) It is obvious that this last equation, XXI., connects the distance, r, with the time, t, as the formula XV. connects the same distance r with the true anomaly, v; that is, with the angular elongation in the orbit, from the position of least distance. But it would be improper here to delay on any of the elementary conse quences of these two known equations: although it seemed useful to show, as above, how the equations themselves might easily be deduced by quaternions, and be connected with the theory of the hodograph.

(16.) The equation II. may be interpreted as expressing, that the parallelogram (comp. Fig. 32) under the vectors a and Da of position and velocity, or under any two corresponding vectors (5.) of the orbit and hodograph, has a constant plane and area, represented by the constant vector ẞ, which is perpendicular (1.) to that plane. But it is to be observed that, by (2.), these constancies, and this representation, are not peculiar to the law of the inverse square, but exist for all other laws of central force.

(17.) In general, if any scalar function R (instead of Mr-2) represent the accelerating force of attraction, at the distance r from the fixed centre o, the differential equation of motion will be (instead of I.),

and if we still write VaDa = ẞ, as in II., the formula IV. will give,

D3a

XXIII. . . D3a - DR. Ua - Rr-2ẞUa, and XXIV... V

D'a

(18.) Applying then the general formula 414, I., we have, for any law* of force,

The general value XXVI., of the radius of curvature of the hodograph, was geometrically deduced in the Paper of 1846, referred to in a recent Note.

CHAP. III. PRODUCT OF OPP. VELOCITIES, POTENTIAL.

721

of which the last not only conducts, in a new way, for the law of nature, to the constant value (7.), h = Mc, but also proves, as stated in (6.), that for any other luw of central force the hodograph cannot be a circle, unless indeed the orbit happens to be such, and to have moreover the centre of force at its centre.

(19.) Confining ourselves however at present to the law of the inverse square, and writing for abridgment (comp. (5.)),

XXVII. . . K = OH = εy = Vector of Centre H of Hodograph,

which gives, by (5.) and (7.),

XXVIII... TK = eh,

the origin o of vectors being still the centre of force, we see by the properties of the circle, that the product of any two opposite velocities in the orbit is constant; and that this constant product* may be expressed as follows,

XXIX... (e-1) hŪk. (e + 1) hÚr = h2 (1 − e2) = Ma ̄1,

by XVIII, and XX.

-

(20.) The expression XXIX. may be otherwise written as x2 y2; and if v be the vector of any point u external to the circle, but in its plane, and u the length of a tangent UT from that point, we have the analogous formula,

XXX... u2 = y2 — (v − k )2 = T (v − k )2 — h2.

(21.) Let and be the vectors or, or' of the two points of contact of tangents thus drawn to the hodograph, from an external point u in its plane; then each must satisfy the system of the three following scalar equations,

XXXI... Syr = 0; XXXII... (7 − x)2 = y2; XXXIII. . . S (7 − k) (v − k) = y2 ; whereof the first alone represents the plane; the two first jointly represent (comp. (5.)) the circle; and the third expresses the condition of conjugation of the points T and u, and may be regarded as the scalar equation of the polar of the latter point. It is understood that Syv = 0, as well as Syk = 0, &c., because y is perpendicular (3.) to the plane.

(22.) Solving this system of equations (21.), we find the two expressions, XXXIV. . . T = x + y (y + u) (v − x) ̄1; XXXIV'. . . r' = k + y ( y − a) (v − k) ̃1 ; in which the scalar u has the same value as in (20.). As a verification, these expressions give, by what precedes,

=

* In strictness, it is only for a closed orbit, that is, for the case (8.) of the centre of force being interior to the hodograph (e <1), that two velocities can be opposite ; their vectors having then, by the fundamental rules of quaternions, a scalar and positive product, which is here found to be Ma1, by XXIX., in consistency with the known theory of elliptic motion. The result however admits of an interpretation, in other cases also. It is obvious that when the centre o of force is exterior to the hodograph, the polar of that point divides the circle into two parts, whereof one is concave, and the other convex, towards o; and there is no difficulty in seeing, that the former part corresponds to the branch of an hyperbolic orbit, which can be described under the influence of an attracting force: while the latter part answers to that other branch of the same complete hyperbola, whereof the description would require the force to be repulsive.

and

XXXV... S(T−k) (T− v) = 0; XXXV'... S(7′ — k) (7′ — v) = 0;
XXXVI. . . (r — v)2 = (†′ — v)2 = — u3.

In fact it is found that

XXXVII. . T-2

and

- v = u(u + y) (v − k)`1 ; XXXVIII... T(u + y) = T (v − x); XXXIX. . . (TM – v) (7 − k) = uy ;

u+y being here a quaternion.

(23.) If v be the vector ov' of any point v', on the polar of the point u with respect to the circle, then changing 7 to v', and u to z, in XXXIV., we find this rector form (comp. (21.)) of the equation of that polar,

XL... v' = x + y (y + z) (v − x) ́1,

or, by an easy transformation,

XLI... (h2 + u2) v′ = h3v + u2x +zy (k − v),

in which z is an arbitrary scalar.

(24.) If then we suppose that u' is the intersection of the chord TT with the right line ou, the condition

but

XLII... Vo'v=0 will give XLIII... zy v2 – Sky'

the coefficient then of к, in the expanded expression for v', disappears as it ought to do: and we find, after a few reductions,

a result which might have been otherwise obtained, by eliminating a new scalar y between the two equations,

XLVI... v=yv, S(yv−k) (v − k) = y2.

(25.) Introducing then two auxiliary vectors, λ, μ, such that

XLVII. . . λ = v1Skv, or SKV = vλ = λv,

and therefore XLVII'. . . λ − x = v1Vкv, Skλ=λ3, (A − x)2 = x2 — X3,

and

-K

we have the very simple relation,

XLIX... (v-λ) (v' —λ) = (μ −λ), or

whence μ, (μ-x)2=y2,

L... LU. LU′ = LM2,

if λ = OL, and μ=OM. Accordingly, the point L is the foot of the perpendicular let fall from the centre II on the right line ou, while м is one of the two points м, M' of intersection of that line with the circle; so that the equation L. expresses, that the points u, u' are harmonically conjugate, with respect to the chord MM', of which L is the middle point, as is otherwise evident from geometry.

(26.) The vector a of the orbit (or of position), which corresponds to the vector (= Da) of the hodograph (or of velocity), and of which the length is Ta=r= the distance, may be deduced from by the equations,

LI... a = r (k − r) y', and LII... Vra = -ẞ= My-1;

whence follow the expressions,

LIII... Potential =

=

Mr-1 (say) P= ST (k − T) = Sv (k − T);

CHAP. III.] CONSTRUCTIONS for the POTENTIAL.

723

the second expression for P being deduced from the first, by means of the relation XXXV.

(27.) The first expression LIII. for P shows that the potential is equal, Ist, to the rectangle under the radius of the hodograph, and the perpendicular from the centre o of force, on the tangent at T to that circle; and IInd, to the square of the tangent from the same point T of the hodograph, to what may be called the Circle of Excentricity, namely to that new circle which has оH for a diameter. And the first of these values of the potential may be otherwise deduced from the equality (7.) of the mass M, to the product he of the radius h of the hodograph, multiplied by the constant c of double areal velocity, or by the constant parallelogram (16.) under any two corresponding vectors.

(28.) The second expression LIII. for the potential P, corresponding to the point T of the hodograph, may (by XXXIV., &c.) be thus transformed, with the help of a few reductions of the same kind as those recently employed:

q being thus an auxiliary quaternion; and in like manner, for the other point T' lately considered, we have the analogous value,

(29.) In fact, the same second expression LIII. shows, that if v and v' be the feet of perpendiculars from T and T′ on HL, then the potentials are,

LXI... P=OU.TV, and P'=OU. T'V' ;

and it is easy to prove, geometrically, that the segment v'L is the harmonic mean between what may be called the ordinates, TV, T'v', to the hodographic axis HL.

(30.) If we suppose the point u to take any new but near position u, in the plane, the polar chord TT', and (in general) the length u of the tangent UT, will change; and we shall have the differential relations:

and

LXII... dr = (7 − v) ̄1S (7 − k) dv ;

LXII'. . . dr' = (r' — v) ́1S (r' — k) dv;
LXIII... du=u ̄1S (k − v) dv.

(31.) Conceiving next that u moves along the line ou, or LU, so that we may