Analytically thus (Thomson and Tait, Nat. Phil.}:— Let x, y, z be the coordinates of P in the orbit, ξ, ∙η, ζ those of the corresponding point T in the hodograph, then t=⅛ w =⅛ —

ς dt ’ 1 dt , * dt ’ therefore dξ - ⅜ ∏∖

<Γ⅛^d 2 y dt i dt i dt 2

Also, if s be the arc of the hodograph, ^-√ θ) ι <

· *

Equation (1) shows that the tangent to the hodograph is parallel to the line of resultant acceleration, and (2) that the velocity in the hodograph is equal to the acceleration.

Every orbit must clearly have a hodograph, and, conversely, every hodograph a corresponding orbit; and, theoretically speaking, it is possible to deduce the one from the other, having given the other circumstances of the motion. We give a few examples :—

1. For uniform motion in a straight line the hodograph is easily seen to be a point.

2. For uniform or variable acceleration in a straight line the hodograph is the line described by a point moving with uniform or variable velocity.

3. For uniform circular motion the hodograph is also a circle. In this case it may be useful to show how the conception of the hodograph leads easily to the ordinary expression for the so-called centrifugal force.

Let P (fig. 2) describe the circumference PP 1 P 2 with uniform velocity v, and from the centre 0 draw OT, OT 1 , OT 2 , &c., equal to each other and parallel to the tangents at P, P 1 . P 2 respectively, then TT 1 T 2 is the hodograph circle. Also let a equal the acceleration of P, which also equals the velocity of T; then, since T describes the hodograph uniformly in the same time that P describes the orbit, we have v OP r . _ v 2

α~^0T - V ' ‘ a ~T' It is evident that the tangent at T is parallel to PO the direction of acceleration at P.

4 . For simple harmonic motion the hodograph is also simple harmonic motion, and similarly for elliptic harmonic motion.

For the former we have s=a cos (nt + e); .∙. ν≈s≈ -na sin (πZ + e) = nα cos (ni+e + ⅜π); which indicates simple harmonic motion with changed amplitude and phase.

5. For parabolic motion the hodograph is a straight line.

Let OF (fig. 3) be the velocity of projection. Resolve it vertically and horizontally; the horizontal component OH is constant, so that the hodograph must be the vertical line FHT. The velocity at any point P of the parabola is evidently represented by the line OT drawn parallel to the tangent at P.

Analytically thus :—If x, y be the coordinates of the moving point and ξ and ∙η those of the hodograph, we have x=0 , ÿ = -/(/being the vertical acceleration);

.∙∙ ξ=0, ή =-/;

. ∙. ξ-c, τj = c'-∕i, which indicates a vertical straight line described with uniform velocity ∕.

6. For central forces the hodograph will vary with the law of acceleration towards the centre. Hamilton showed that, for the Newtonian law of the inverse square of the distance, the hodograph is always a circle, and for that reason he designated that law the law of the circular hodograph.

Assuming the centre of force as origin for the hodograph, we see that, from definition, the tangent and radius vector at any point P of the orbit are respectively parallel to the radius vector and tangent at the corresponding point T of the hodograph (fig. 4). These four lines thus enclose a parallelogram PT, whose shape is constantly changing, although its area is constant, being equal to the constant rectangle YT contained by the velocity and the perpendicular on the tangent. As usual, let TY=φ=Λ. Also the angle between two consecutive tangents to the hodograph is equal to the angle between the corresponding radii vectores of the orbit, and hence/if 0 be the angle between OP and some initial line, and s the arc of the hodograph, the ordinary formula for curvature gives

1 dθ _ 0

ρ ds ~ s ’ but 7 t =r 2 0, and = velocity in hodograph = acceleration in orbit=P (say);

• 1 p = P i p r 2

∙∙∙ p =-F

a most useful expression for the radius of curvature of the hodograph to any central orbit.

If the acceleration vary inversely as the square of the distance, P = - 2 - > where M is the mass at the centre acting upon unit mass in the orbit. Substituting in (3), we get

M . . p = — = constant.

h

Hence, for this law of force, and evidently for it only, the hodograph is a circle.

Assuming that the hodograph is a circle, we can show that the orbit for this law of force must be a conic section. Let CTB (fig. 5) be the hodograph circle, 0 the origin, and H the centre; and let the ratio = e . Draw OP parallel to the tangent at T and, therefore, perpendicular to TG. Take OP such that the parallelogram contained by OP and OT = a constant ≈h. Draw OA perpendicular to CB; and let POA = 0; then,

OP.TG=⅛j

.∙. OP. (TH + HG) = ½J butTH= i >=^, and HG = HO cos 0 = eρ cos 0=-v-cos 0;

Γb

·· 0p fv + e⅞cos0)=Λ;

∖ n> h ∕

• OP=r=-· 1

Μ 1 + e cos 0 Hence the locus of P is a conic section whose semi-parameter is

. If e<l> o is within the circle and. the orbit is an ellipse; if e = l, 0 is on the circumference and the orbit is a parabola; and if e >l, 0 is without the circle and the orbit is an hyperbola.

Two values of the potential, V, can readily be found from the above :—

xτ M . . h .

V = —; but r== 7s ; r 1G

M

.∙. V=∙^. TG=TH . TG.

h

Also, since OGH is a right angle, V = TH . TG = the square of the tangent from T to the circle described on HO as diameter.

A beautiful result connected with the hodograph, and one which has attracted the attention of several of the ablest mathematicians, was communicated by Sir William Hamilton to the Royal Irish Academy in March 1847. It is called the theorem of hodographic isochronism, and is thus stated:—If two circular hodographs, having a common chord, which passes through, or tends towards, a common centre of force, be cut perpendicularly by a third circle, the times of hodographically describing the intercepted arcs will be equal. A purely quaternion proof is given of this theorem by Hamilton in his Elements, and, following the hints given by that method, he has also indicated the following geometrical proof.

Let TMT'M ’, WMW'M' (fig. 6), be the two hodographic circles with centres H and L and common chord MM'. Let P, P', on the common chord produced, be the centres of two circles WTW'T' and BAB ’ A', near each other, which cut the hodographs orthogonally. Let 0 be the centre of force, OZ perpendicular to the tangent at T, and TR, T'R' perpendicular to SX. Also let AT mean the arc AT, and similarly for the other small arcs. Draw PY perpendicular to PT.

From similar triangles THΛ and TPY we have

AT _ PY.

TH “PT ’

also from similar triangles THR and PP ’ Y

TH PU . AT PP' .

TR “PY ' ‘ TH. PT ’

.∙.ΛT-∑⅛S.

Similarly, AT” - --^ T ' R '.

If „ and i' be the times of hodographically describing the arcs AT and A'T' respectively,

. AT M . AT M . AT ... , r . .. . .. 1 . τ , λ “mF 2 " m 2 F*^ " ~Ψ~ (lf v be the p° tentlal at τ >∙

But V = HT .OZ = OP. TR from similar triangles OPZ and THR;

. f M . AT M . PP'. TR Μ. PU

OP 2 . TR 2 OP 2 . TR 3 . PT - OP 3 . PT . TR Similarly, f , _ M . PU. T z R' M . PP'

“OP 2 . T'R' 2 . PT “OP 2 . PT . T'R' ’

I + ι' - M . P P' ( J_ 1 ∣ _ 2M . PP' I TR + T ’ R') OP 2 . PT. ( TR + T'R'1 OU. PT ( 2TR . T'R' I' 2TB T'R'

Now ιi ^, = the harmonic mean between TR and T'R' = QL; • t + t' - 2M rp '

OP 2 . PT . QL ’

From this expression we see that the time of describing the twβ small arcs TA and T'A' is independent of the radius of the hodograph and the distance of its centre from L. Hence it is equal to the time of describing the two arcs BW and B'w. By continuing the process of drawing orthogonale we arrive at the conclusion that the time of describing the whole arc ATT'A' is equal to the time of describing the arc BWW'B'. A very simple analytical proof is given in Tait and Steele’s Dynamics of a Particle. Others not so simple by Cayley and Droop are to be found in the Quarterly Journal of Mathematics. We can only mention that the theorem of Lambert can be deduced from the above theorem of hodographic isochronism without using any property of conic sections.

Sir William Hamilton has observed that we have a good instance of a hodograph in the curve of aberration of a star, which is merely the hodograph of the earth’s annual motion. The fact that this curve is a circle in a plane parallel to the earth’s orbit, abstracted, however, from the general idea of the hodograph, was known long before the date of the hodograph. It will be found clearly stated and proved in Woodhouse’s Astronomy, of date 1821, and from allusions there it appears to have been known even earlier. As an application of the hodograph, Thomson and Tait point out that the heat and light received by a planet from the sun in any time are proportional to the corresponding arc of the hodograph.

See Proc. Roy. Irish Acad., 1846; Hamilton’s Elements of Quaternions·, Tait, Proc. R. S. E., 1867-8; Tait’s Quaternions; Thomson and Tait, Nat. Phil.