CHAP. III.] LAMBERT'S THEOREM, PARTIAL DIFF. EQUATIONS. 729

integrals in (39.), that the time of describing an arc PP′ of an orbit, with the law of the inverse square, is a function (comp. (36.)) of the three ratios,

which is a form of Lambert's Theorem, but presents itself here as deduced from the recently stated Theorem of Hodographic Isochronism (38.), without the employment of any property of conic sections.

(48.) The differential equation I. of the present relative motion may be thus written (comp. 418, I., and generally the preceding Series 418):

CIV... S. Dada+dP=0, whence CV... T= P+ H,

as in 418, X., if we now write,

(49.) Integrating CIV. by parts, &c., and writing (as in 418, XII. XXII.),

so that F may again be called the Principal Function and V the Characteristic Function of the motion, we have the variations,

CXI... &F=Srda - Sr'da' – Hồt;

CXII. . . &V=Srda - Sr'da' + tô H;

in which a, a' (instead of ao, a) denote now what may be called the initial and final vectors (OP, OP') of the orbit; whence follow the partial derivatives,

CXIII... DaF=DaV=T;

CXIV... (D,F) = − H ;

CXIII'... DaF= Da'V=- T';
and CXV... DнV=t;

F being here a scalar function of a, a', t, while V is a scalar function of a, a', H, if M be treated as given.

(50.) The two vectors a, a' can enter into these two scalar functions, only through their dependent scalars r, r', s (comp. 418, (17.)); but

CXVI... dr = -r ̄1Sada, dr'=-r′-1Sa'da', ds=-s ̄1S (a′ − a) (du' – da); confining ourselves then, for the moment, to the function V, and observing that we have by CXII. the formula,

CXVII... S(rda - r'da') = D, V. dr + Dr V. dr' + D ̧ V. ds, in which the variations da, da' are arbitrary, we find the expressions,

CXVIII. . . T = — ar‍1D, V + (a' − a) s ̄1D, V ;

permitted to conceive the motion to be performed along either of the two elliptic arcs, PP, PP, which together make up the whole periphery. But into details of this kind we cannot enter here.

CXXI... rr + r'r' = (r + r') v' || v || a'-a,

the chord TT' of the hodograph, in Figures 86, 87, being divided at u' into segments TU', U'T', which are inversely as the distances r, r', or as the lines OP, OP' in the orbit; we have therefore the partial differential equation,

CXXII... D,V=D, V, and similarly, CXXIII... D‚F=D,F;

so that each of the two functions, F and V, depends on the distances r, r', only by depending on their sum, r + r'.

(52.) Hence, if for greater generality we now treat M as variable, the Principal Function F, and therefore by CXIV. its partial derivative H=- (D,F), are functions of the four scalars,

CXXIV. . . r + r', s, t, and M.

(53.) And in like manner, the Characteristic Function (or Action-Function) V, and its partial derivative (by CXV.) the Time, t=DV, may be considered as functions of this other system of four scalars (comp. (47.)),

CXXV...r+r', s, H, and M;

no knowledge whatever being here assumed, of the form or properties of the orbit, but only of the law of attraction.

(54.) But this dependence of the time, t, on the four scalars CXXV., is a new form of Lambert's Theorem (47.); which celebrated theorem is thus obtained in a new way, by the foregoing quaternion analysis.

(55.) Squaring the equations CXVIII. CXVIII'., attending to the relation CXXII., and changing signs, we get these new partial differential equations,

Hence, by merely algebraical combinations (because P= Mr, and P′ = Mr'-1), we

CHAP. III.] DEFINITE INTEGRALS FOR ACTION AND TIME. 731

(56) But, by CXII. CXVII. CXXII., we have the variation,

CXXXI...d V-td H = { (D, V+ D ̧ V) ♪ (r + r2 + s ) + † (D, V− D ̧V) & (r + r' − s) ;

and the function V vanishes with t, and therefore with s, at least at the commencement of the motion; whence it is easy to infer the expressions,*

As a verification,† when t and s are small, and therefore r' nearly=r, we have thus the approximate values,

CXXXIV... V = (2P+ 2H) s = (27)1s = 2Tt;

CXXXV. t= (2P+ 2H) ̄1s = (2 T)1s ;

in which s may be considered to be a small arc of the orbit, and (2T) the velocity with which that arc is described.

(57.) Some not inelegant constructions, deduced from the theory of the hodograph, might be assigned for the case of a closed orbit, to represent the excentric and mean anomalies; but whether the orbit be closed or not, the arc TMT of the hodographic circle, in Fig. 86, represents the arc of true anomaly described: for it subtends at the hodographic centre и an angle THT', which is equal to the angular motion POP' in the orbit.

(58.) We may add that, whatever the special form of the orbit may be, the equations CXVIII. CXVIII'. give, by CXXII.,

CXXXVI... 7' -
-T= = (Ua' + Ua) D, V;

from which it follows that the chord TT of the hodograph is parallel to the bisector of the angle POP' in the orbit: and therefore, by XCI., that this angle is bisected by oq in Fig. 87, so that the segments PR, RP', in that Figure, of the chord PP' of the orbit, are inversely proportional to the segments ru', U'T′ of the chord TT' of the hodograph.

(59.) We arrive then thus, in a new way, and as a new verification, at this known theorem: that if two tangents (QP, QP′) to a conic section be drawn from

* Expressions by definite integrals equivalent to these, for the action and time in the relative motion of a binary system, were deduced by the present writer, but by an entirely different analysis, in the First Essay, &c., already cited, and will be found in the Phil. Trans. for 1834, Part II., pages 285, 286. It is supposed that the radical in CXXXIII. does not become infinite within the extent of the integration; if it did so become, transformations would be required, on which we cannot enter here.

+ An analogous verification may be applied to the definite integral LXXXI.; in which however it is to be observed that both r+r′ and s vary, along with the variable w: whereas, in the recent integrals CXXXII. CXXXIII., r + r' is treated as constant.

any common point (Q), they subtend equal angles at a focus (0), whatever the special form of the conic may be.

(60.) And although, in several of the preceding sub-articles, geometrical constructions have been used only to illustrate (and so to confirm, if confirmation were needed) results derived through calculation with quaternions; yet the eminently suggestive nature of the present Calculus enables us, in this as in many other questions, to dispense with its own processes, when once they have indicated a definite train of geometrical investigation, to serve as their substitute.

(61.) Thus, after having in any manner been led to perceive that, for the motion above considered, the hodograph is a circle* (5.), of which the radius HT is equal (7.) to the attracting mass M, divided by the constant parallelogram (16.) under the vectors op, or of position and velocity, in the recent Figures 86 and 87, which parallelogram is equal to the rectangle under the distance or in the orbit, and the perpendicular oz let fall from the centre o of force on the tangent Ur to the hodograph, we see geometrically that the potential P, or the mass divided by the distance, for the point P of the orbit corresponding to the point r of the hodograph, is equal (as in (27.)) to the rectangle under HT and oz, and therefore, by the similar triangles HTV, UOz, to the rectangle under ou and Tv (as in (29.)).

(62.) In like manner, the three potentials corresponding to the second point r' of the first hodograph, and to the points w and w' of the second hodograph, in Fig. 86, are respectively equal to the rectangles under the same line ou, and the three other perpendiculars T'v', wx, w'x', on what we have called (29.) the hodographie axis, HL; so that, for these two pairs of points, in which the two circular hodographs, with a common chord MM', are cut by a common orthogonal with u for centre, the four potentials are directly proportional to the four hodographic ordinates (29.).

(63.) And because the force (Mr-2) is equal to the square of the potential (Mr-1), divided by the mass (M), the four forces are directly as the squares of the four ordinates corresponding; each force, when divided by the square of the corresponding hodographic ordinate, giving the constant or common quotient,

CXXXVII... OU2: M.

(64.) It has been already seen (31.) to be a geometrical consequence of the two pairs of similar triangles, NTT,, NT'T', and NTV, NT'v', that the two small ares of the first hodograph, near T and T', intercepted between two near secants from the pole N of the fixed chord MM', or between two near orthogonal circles, with u aud u, for centres, are proportional to the two ordinates, TV, T'V'.

(65.) Accordingly, if we draw, as in Fig. 86, the near radius (represented by a

This follows, among other ways, from the general value XXVI. for the radius of curvature of the hodograph, with any law of central force; which value was geometrically deduced, as stated in the Note to page 720, compare the Note to page 719, by the present writer, in a Paper read before the Royal Irish Academy in 1846, and published in their Proceedings. In fact, that general expression for the radius of hodographic curvature may be obtained with great facility, by dividing the element fat of the hodograph (in which ƒ denotes the force), by the corresponding element er dt of angular motion in the orbit.

CHAP. III.] DISTANCE OF COMET OR PLANET FROM EARTH. 733

dotted line from H) of the first hodograph, and also the small perpendicular UY, erected at the centre u of the first orthogonal to the tangent UT, and terminated in y by the tangent from the near centre U,, the two new pairs of similar triangles, THT,, UTY, and THV, UU,Y, give the proportion,

CXXXVIII... TT,: TV = UU,: UT;

which not merely confirms what has just been stated (64.), for the case of the first hodograph, but proves that the four small arcs, of the two circular hodographs in Fig. 86, intercepted between the two near orthogonals, are directly proportional to the four ordinates already mentioned.

(66.) But the time of describing any small hodographic arc is the quotient (32.) of that are divided by the force; and therefore, by (63.), (65.), the four small times are inversely proportional to the four ordinates. And the harmonic mean U'L between the two ordinates TV, T'v' of the first hodograph, does not vary when we pass to the second, or to any other hodograph, with the same fixed chord MM', and the same orthogonal circles; it follows then, geometrically, that the sum (33.) of the two small times is the same, in any one hodograph as in any other, under the conditions above supposed: and that this sum is equal to the expression,

CXXXIX...

2 M.UU' OU2. UT.U'L

which agrees with the formula LXXIII.

==

2 M. UU. UL
OU. LM2. UT

(67.) On the whole, then, it is found that the Theorem of Hodographic Isochronism (38.) admits of being geometrically* proved, although by processes suggerted (60.) by quaternions: and sufficient hints have been already given, in connexion with Figure 87, as regards the geometrical passage from that theorem to the wellknown Theorem of Lambert, without necessarily employing any property of conic sections.

420. As a fifth specimen, we shall deduce by quaternions an equation, which is adapted to assist in the determination of the distance of a comet, or new planet, from the earth.

(1.) Let M be the mass of the sun, or (somewhat more exactly) the sum of the masses of sun and earth; and let a and w be the heliocentric vectors of earth and comet. Write also,

I... Ta=r, Tw=w, T(w-a)=z, U(w− a)= p,

so that r and w are the distances of earth and comet from the sun, while z is their distance from each other, and p is the unit-vector, directed from earth to comet. Then (comp. 419, I.),

*It appears from an unprinted memorandum, to have been nearly thus that the author orally deduced the theorem, in his communication of March, 1847, to the Royal Irish Academy; although, as usually happens in cases of invention, his own previous processes of investigation had involved principles and methods, of a much less simple character.