a result easily extended, as above. If the law of attraction were supposed different, there would be no difficulty in modifying the expression for the potential accordingly.

(2.) In general, when a scalar, ƒ (as here P), is a function of one or more vec. tors, a, a,... its variation (or differential) can be expressed as a linear and scalar function of their variations (or differentials), of the form Sßda + S3'ĉa' +.... (or ESẞda); in which B, 3'... are certain new and finite vectors, and are themselves generally functions of a, a', ..., derived from the given scalar function ƒ. And we shall find it convenient to extend the Notation† of Derivatives, so as to denote these derived vectors B, B', &c., by the symbols, Daf, Darf, &c. In this manner we shall be able to write,

V... PES (DaP.da);

and the differential equations of motion of the bodies m, m', m",.. will take by I. the forms:

VI... mDa + DaP=0, m'Di2a′ + Da' P = 0, &c.;

(3.) The laws of the centre of gravity, of areas, and of living force, result immediately from these equations, under the forms,

and

VIII... Em Dia = ß;

IX... EmVaD1α = y;

X... T-2m (Dta)2 = P + H ;

in which ẞ, y are constant vectors, H is a constant scalar, and 2T is the living force of the system (comp. 417, (5.)).

(4.) One mode (comp. 417, (2.)) of deducing the three equations, of which these are the first integrals, is the following. To obtain VIII., change every variation da in I. to one common but arbitrary infinitesimal vector, ɛ. For IX., change da to Via, da' to Via', &c.; being another arbitrary and infinitesimal vector. Finally, to arrive at X., change variations to differentials (da to da, &c.), and integrate once, as for the two former equations, with respect to the time t.

(5.) The formula I. admits of being integrated by parts, without any restric tion on the variations da, by means of the general transformation,

XI... S (Dia.da) = DiS (Dra. da) – 4d. (Dɛa)2,

combined with the introduction of the following definite integral (comp. X.),

*It may not be useless here to compare the expression in page 417, for the differential of a proximity.

In this extended notation, such a formula as dfp = 2Svdp would give,

v = Dpfp.

CHAP. III.]

INTERMEDIATE AND FINAL INTEGRALS.

715

(6.) In fact, if we denote by ao, a'o,.. the initial values of the vectors a, a', .. or their values when t = 0, and by Doa, Doa',.. the corresponding values of Dia, Dia',.., we shall thus have, as a first integral of the equation I., the formula, XIII. EmS (Dra. da- Doa. dao) + &F = 0;

in which no variation ôt is assigned to t, and which conducts to important conse

quences.

(7.) To draw from it some of these, we may observe that if the masses m, m',.. be treated as constant and known, the complete integrals of the equations VI. or VII. must be conceived to give what may be called the final vectors of position a, a',.. and of velocity Dia, Dia',...in terms of the initial vectors ao, a'o,.. Doa, Doa',.. and of the time, t: whence, conversely, we may conceive the initial vectors of velocity to be expressible as functions of the initial and final vectors of position, and of the time. In this way, then, we are led to consider P, T, and F as being scalar functions (whether we are or are not prepared to express them as such), of a, a',.. ao, a'o,.. and t; and thus, by (2.), the recent formula XIII. breaks up into the two following systems of equations:

and

XIV. . . mDia + DaF=0,
XV. . . - mDọa + DaF=0,

m'Dia' + Da'F' = 0, &c. ; -m'Doa' + Da'F= 0, &c. ;

whereof the former may be said to be intermediate integrals, and the latter to be final integrals, of the differential equations of motion of the system, which are included in the formula I.

on

(8.) In fact, the equations XIV. do not involve the final vectors of acceleration Da,.. as the differential equations VI. or VII. had done; and the equations XV. express, at least theoretically, the dependence of the final vectors of position a,. the time, t, and on the initial vectors of position ao,.. and of velocity Doa,.. as by (7.) the complete integrals ought to do. And on account of these and other important properties, the function here denoted by F may be called the Principal* Function of Motion of the System.

(9.) If the initial vectors ao,.. and Doa,.. be given, that is, if we consider the actual in space of the mutually attracting system of masses m, from one progress set of positions to another, then the function F' depends upon the time alone; and by its definition XII., its rate or velocity of increase, or its total derivative with respect to t, is thus expressed,

XVI... D1F= P + T.

(10.) But we may inquire what is the partial derivative, say (DF), of the same definite integral F, when regarded (7.) as a function of the final and initial vectors of position a,.. ao,.. which involves also the time explicitly, and is now to be derivated with respect only to that variable t, as if the final vectors a,.. were constant : whereas in fact those vectors alter with the time, in the course of any actual motions of the system.

*This function was in fact so called, in two Essays by the present writer, "On a General Method in Dynamics," published in the Philosophical Transactions (London), for the years 1834 and 1835; although of course coordinates, and not quaternions, were then employed, the latter not having been discovered until 1843: and the notation S, since adopted for scalar, was then used instead of F.

(11.) For this purpose, it is sufficient to observe that the part of the total derivative DF, which arises from the last mentioned changes of a,.. is (by XIV. and X.),

XVII... ES (DF. D1α) = 2 T ;

and therefore (by XVI. and X.), that the remaining part must be,

XVIII. . . (D2F) = P – T = − H.

(12.) The complete variation of the function F is therefore (comp. XIII), when t as well as a,.. and ao, is treated as varying,

XIX... ¿F=- Hdt – EmSD;aða + ΣmSD ̧aðão.

(13.) And hence, with the help of the equations X. XIV. XV., it is easy to infer that the principal function F must satisfy the two following Partial Differential Equations in Quaternions :

so that represents what is called the Action, or the accumulated living force, of the system during the time t, then by X. and XII. the two definite integrals F and I are connected by the very simple relation,

XXIII... V=F+tH;

whence by XIX. the complete variation of V, considered as a function of the final and initial vectors of position, and of the constant H of living force, which does not explicitly involve the time, may be thus expressed,

XXIV... &V=t&H - EmSD,aða + EmSD ̧aða.

(15.) The partial derivatives of this new function V, which is for some purposes more useful than F, and may be called, by way of distinction from it, the Characteristic Function of the motion of the system, are therefore,

(16.) The intermediate integrals (7.) of the differential equations of motion, which were before expressed by the formula XIV., may now, somewhat less simply, be regarded as the result of the elimination of H between the formula XXV. XXVII. ; and the final integrals of those equations VI. or VII., which were expressed by XV., are now to be obtained by eliminating the same constant H between the recent equations XXVI. XXVII.

*The Action, V, was in fact so called, in the two Essays mentioned in the preceding Note. The properties of this Characteristic Function had been perceived by the writer, before those of that which he came afterwards to call the Principal Function, as above.

CHAP. III.] PRINCIPAL AND CHARACTERISTIC FUNCTIONS. 717

(17.) The Characteristic Function, V, is obliged (comp. (13.)) to satisfy the two following partial differential equations,

XXVIII... m ̄1 (Da V)2+P+H=0;

XXIX. . . †Σm ̄1(DaoV)2+ Po+H= 0 ;

it vanishes, like F, when t=0, at which epoch a=aq, a' = a'o, &c.; each of these two functions, F and V, depends symmetrically on the initial and final vectors of position and each does so, only by depending on the mutual configuration of all those initial and final positions.

(18.) It follows (comp. (4.), see also 416, (17.), and 417, (2.)), that the function F must satisfy the two conditions,

XXX... Σ(DaF+ Daol

DaoF) = 0 ;

XXXI... EV (aDaF +α ̧DaoF) = 0 ;

which accordingly are forms, by XIV. XV., of the equations VIII. and IX., and therefore are expressions for the law of motion of the centre of gravity, and the law of description of areas. And, in like manner, the function V is obliged to satisfy these two analogous conditions,

XXXII... Σ(DaV + DaoV)=0 ;

XXXIII. . . ΣV (aDa V + æ ̧Ð ̧ ̧ V) = 0 ;

a0

which accordingly, by XXV. XXVI., are new forms of the same equations VIII. IX., and consequently are new expressions of the same two laws.

(19.) All the foregoing conditions are satisfied when t is small, that is, when the time of motion of the system is short, by the following approximate expressions for the functions Fand V, with the respectively derived and mutually connected expressions for H and t:

419. As a fourth specimen, we shall take the case of a free point or particle, attracted to a fixed centre* o, from which its variable vector is a, with an accelerating force = Mr2, if r = Ta= the distance

*When two free masses, m and m', with variable vectors a and a', attract each other according to the law of the inverse square, the differential equation of the relative motion of m about m' is, by 418, VII.,

I'. . . D2 (a — a') = (m + m') (a − a′)-1r 1, if

1= T(a-a');

and this equation I'. reduces itself to I., when we write a for a-a', and M for m + m'.

of the point from the centre, while M is the attracting mass: the differential equation of the motion being,

I. . . D2a = Ma ̄1r1,

if D (abridged from D,) be the sign of derivation, with respect to the time t.

(1.) Operating on I. with V.a, and integrating, we obtain immediately the equation (comp. 338, (5.)),

II... VaDaß = const. ;

which expresses at once that the orbit is plane, and also that the area described in it is proportional to the time; Uß being the fixed unit-normal to the plane, round which the point, in its angular motion, revolves positively; and Tẞ representing in quantity the double areal velocity, which is often denoted by c.

(2.) And it is important to remark, that these conclusions (1.) would have been obtained by the same analysis, if r-1 in I. had been replaced by any other scalar function, f(r), of the distance; that is, for any other law of central force, instead of the law of the inverse square.

(3.) In general, we have the transformation,

III... a 1Ta1 dUa: Vada,

because, by 334, XV., &c., we have,

IV... dUa = V (da. a ̄1).Ua = a ̄2Ua.Vada = a ̄1Ta‍1. Vada;

the equation I. may therefore by II. be transformed as follows,

and thus it gives, by an immediate integration,

VII... Da = y (Ua − e), or VII'... Da = (ε − Ua) y,

being a new constant vector, but one situated in the plane of the orbit, to which plane ẞ and y are perpendicular.

(4.) But a, Da, D2a are here (comp. 100, (5.) (6.) (7.)) the vectors of position, velocity, and acceleration of the moving point; and it has been defined (100, (3.)) that if, for any motion of a point, the vectors of velocity be set off from any common origin, the curve on which they terminate is the Hodograph of that motion.

(5.) Hence a and Da, if the latter like the former be drawn from the fixed point o, are the vectors of corresponding points of orbit and hodograph; and becau-e the formula VII. gives,

VIII.

SyDa = 0, and IX... (Da + yε)2 = y2,

it follows that the hodograph is, in the present question, a Circle, in the plane of the

Compare Fig. 32, p. 98; see also pages 100, 515, 578, from the two latter of which it may be perceived, that the conception of the hodegraph admits of some purely geometrical applications.