LXV... dv = (p − λ) dx = v (x − e ́) dx, with x>1>é,

if u be on LM prolonged, and if o be on the concave side of the arc TMT'; and thus, by LIII., the differential expressions (30.) become,

and

LXVI. . . dr = (v − 7) ̄1P(x − e')-1dx; dr'=(v-r') ̄1P′ (x − e')-1dz ;

so that

LXVII... du= u-1Sq. (x - e')-1dx, with Sq = v(λ − v);

Such then are the lengths of the two elementary arcs TT, and T'T,' of the hodograph, intercepted between two near secants NTT' and NT,T,' drawn from the pole N of the chord мм', and having u and u, for their own poles; and we see that these arcs are proportional to the potentials, P and P', or by LXI. to the ordinates, TV, T'v', or finally to the lines NT, NT': and accordingly we have the inverse similarity (comp. 118), of the two small triangles with N for vertex,

(32.) For any motion of a point, however complex, the element dt of time which corresponds to a given element dDa of the hodograph, is found by dividing the latter element by the vector D2a of accelerating force; if then we denote by dt and d the times corresponding to the elements dr and dr' (31.), we have the expressions,

CHAP. III.] THEOREM OF HODOGRAPHIC ISOCHRONISM.

725

because, for the motion here considered, the measure or quantity of the force is, by I. and LIII.,

LXXI... TD2a = Mr2 = M1P2.

(33.) The times of hodographically describing the two small circular ares, T,T and TT, are therefore inversely proportional to the potentials, or directly proportional to the distances in the orbit; and their sum is,

so that the sum of the two small times may be thus expressed,

in which Figure u'w is an ordinate of a semicircle, with the chord мM' of the hodograph for its diameter.

(35.) The two near secants (31.), from the pole N of that chord, have been here supposed to cut the half chord LM itself, as in the cited Figure 86; but if they were to cut the other half chord LM', it is easy to prove that the formula LXXVIII. LXXIX. would still hold good, the only difference being that the angle w, or MLW, would be now obtuse, and its secant <- 1.

(36.) A circle, with u for centre, and u for radius, cuts the hodograph orthogonally in the points T and T'; and in like manner a near circle, with u, for centre, and u+ du for radius, is another orthogonal, cutting the same hodograph in the near points T, and T,' (31.). And by conceiving a series of such orthogonals, and observing that the differential expression LXXVIII. depends only on the four scalars, M-1a3, e', w, and dw, which are all known when the mass M and the five points o, L, M, U, U, are given, so that they do not change when we retain that mass and those points, but alter the radius h of the hodograph, or the perpendicular HL let fall from its centre H on the fixed chord Mм', we see that the sum of the times (comp. (33.), of hodographically describing any two circular arcs, such as TT and T'T', even if they be not small, but intercepted between any two secants from the pole N of the fixed chord, is independent of the radius (h), or of the height HL of the centre н of the hodograph.

(37.) If then two circular hodographs, such as the two in Fig. 86, having a common chord MM', which passes through, or tends towards, a common centre of force o, with a common mass M there situated, be cut by any two common orthogonals, the sum of the two times of hodographically describing (33.) the two intercepted arcs (small or large) will be the same for those two hodographs.

(38.) And as a case of this general result, we have the following Theorem" of Hodographic Isochronism (or Synchronism):

"If two circular hodographs, having a common chord, which passes through, er 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.”

For example, in Fig. 86, we have the equation,

LXXX... Time of TMT = time of WMW'.

(39.) The time of thus describing the arc TмT′ (Fig. 86), if this arc be throughout concavet towards o (so that >1>e', as in LXV.), is expressed (comp.

and the time of describing the remainder of the hodographic circle, if this remaining arc T'M'T be throughout concave towards the centre o of force, is expressed by this other integral,

(40.) Hence, for the case of a closed orbit (e'2 < 1, e<1, a > 0), if a denote the mean angular velocity, we have the formula,

The same result, for the same case of elliptic motion, may be more rapidly obtained, by conceiving the chord MM' through o to be perpendicular to OH; for, in this position of that chord, its middle point L coincides with o, and e' = 0 by LXIV. (41.) In general, by LXXVI., we are at liberty to make the substitution,

supposing then that e'= 1, or placing o at the extremity M' of the chord, we have by LXXXI.,

for, when the centre of force is thus situated on the circumference of the hodographic circle, we have by (8.) the excentricity e = 1, and the orbit becomes by XV. a para

* This Theorem, in which it is understood that the common centre of force (0) is occupied by a common mass (M), was communicated to the Royal Irish Academy on the 16th of March, 1847. (See the Proceedings of that date, Vol. III., page 417.) It has since been treated as a subject of investigation by several able writers, to whom the author cannot hope to do justice on this subject, within the very short space which now remains at his disposal.

Compare the Note to page 721.

CHAP. III.] CHORD of orbit, SUM OF DISTANCES.

727

bola. For hyperbolic motion (e22> 1, e > 1, a <0), the formula LXXXI. (with or without the substitution LXXXV.) is to be employed if e' <-1, that is, if o be on LM' prolonged; and the formula LXXXII., if e'>1, e'<sec w, that is, if o be situated between м and U.

(42.) For any law of central force, if P, P' be the points of the orbit which correspond to the points T, T' of the hodograph, and if q be the point of meeting of the tangents to the orbit at P, P', as in the annexed Figure 87, while the tangents to the hodograph at T, T' meet as before in u, we shall have the parallelisms,

LXXXVIII. . . .. OP=α, OP' a', or = Da=T, OT' = Da' = 7', OU=V, OQ=w, most of which notations have occurred before, we have the equations,

LXXXIX. . . 0 = Va (r− v) = Va' (v − 7') = Vr (w− a) = Vr' (a' — w) ; thus XC... Vav Var = ẞ= Va'r'=Va'v, a' - a || v, PP' || ou, and XCI... VTw= Vra = - ẞ = Vr'a' = Vr'w', T-T' || W, TT || 0Q. Geometrically, the constant parallelogram (16.) under op, or, or under op', or', is equal, by LXXXVII., to each of the four following parallelograms: I. under op, ou; II. under op', ou; III. under oq, or; and IV. under oq, or'; whence PP'|| OU, and TTOQ, as before.

(43.) The parallelism XC. may be otherwise deduced for the law of the inverse square, with recent notations, from the quaternion formulæ,

and which may be obtained in various ways; whence it may also be inferred, that

if s denote the length T (a' — a) of the chord PP' of the orbit, then (comp. Fig. 86),

8

and the lines LT, LT' are therefore in length proportional to the potentials, P, P'; their directions are equally inclined to that of ou, but at opposite sides of it, so that the line LU bisects the angle TLT'. Accordingly (see Fig. 86), the three points T, L, T′ are on the circle (not drawn in the Figure) which has HU for diameter; so that the angles ULT, TLU are equal to each other, as being respectively equal to the angles UTT', TT'U, which the chord TT' of the hodograph makes with the tangents at its extremities: the triangles TLV, T'LV' are therefore similar, and LT is to LT' as TV to T'v', that is, by LXI., as P to P', or as r' to r. (45.) Again, calculation with quaternions gives, (v − T) (\ − T) _ (v — 7′) (A − 7′)

XCVII...

=

such then is the common ratio, of the segments TU', U'T' of the base TT' of the triangle TLT', to the adjacent sides LT, LT', which are to each other as r ́to r (44.); and because this ratio is also that of s to r+r', by (43.), we have the proportion, XCIX... OP: OP': PP'=r: r': s :8= LT: LT: TT',

and the formula of inverse similarity (118),

C... ALT'T x' OPP'.

Accordingly (comp. the two last Figures), the base angles OPP', OP'P of the second triangle are respectively equal, by the parallelisms (42.), to the angles TUL, TUL, and therefore, by the circle (44.), to the base angles TT'L, TTL, of the first triangle: but the two rotations, round o from P to P', and round L from T' to T, are oppositely directed.

(46.) The investigations of the three last subarticles have not assumed any knowledge of the form of the orbit (as elliptic, &c.), but only the law of attraction according to the inverse square, or by (6.) the Law of the Circular Hodograph. And the same general principles give not only the expression LXXVI. for the constant Ma-, but also (by LX. LXIV. LXXIV. LXXIX.) this other expression,

which last may be considered as a quadratic in e', assigning two values (real or imaginary) for that scalar, when the first member of CII. and the angle w are given; the sine of this latter angle being already expressed by XCIII.

(47.) Abstracting, then, from any ambiguity* of solution, we see, by the definite

*That there ought to be some such ambiguity is evident from the consideration, that when a focus o, and two points P, P' of an elliptic orbit are given, it is stil