its partial differential coefficient ($18,0, dr), obtained by treating q as constant,

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vanishes at the limit r=9; nor need it be varied with respect to q, because, by (186.),

M +m x 89

M ge ou

(209.) q it may therefore be treated as constant, and we find at last {7, w} : 0,

(210.) the two terms (199.) or (203.) both tending to infinity when r tends to


but always destroying each other.

36. Collecting now our results, and presenting for greater clearness each combination under the two forms in which it occurs when the order of the elements is changed, we have, for each binary system, the following thirty expressions :

{x,a} = 0, {r, } = 0, {x,y} = 0, {,7} = 0, {,w} = -1,

{, x} = 0, {n, } = 0, {1, v} = 1, {1, r} = 0, {A, w}

= 0,
{pi, x} = 0, {f, a} = 0, {fe, } = 0, {f, T} = 1, {fe, w} = 0,

{v, x} = 0, {v,a} = -1, {v, k} = 0, {v, r} = 0, {v,w} = 0,
a . ,

{T,x} = 0, {1,2} = 0, {T, k} = -1, {t,v} = 0, {7,w} = 0,
} :

{w,x} = 1, {w,} = 0, {w, pe} = 0,{w, w} = 0, {W,7} = 0);

v wr so that the three combinations

{fe, T} {W, %} {1, v} are each equal to positive unity; the three inverse combinations

{T,f} {x, w} {v,a} are each equal to negative unity; and all the others vanish. The six differential equations of the first order, for the 6 varying elements of any one binary system (m, M), are therefore, by (02.), du

, dt



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and, if we still omit the variation of t, they may all be summed up in this form for the variation of H2,

δH, = Σ.m (μ'
à H2= 2.m (ror - g'dir + add x - x'ow ta'du-dda),

(T2.) which single formula enables us to derive all the 6 n-6 differential equations of the first order, for all the varying elements of all the binary systems, from the variation or from the partial differential coefficients of a single quantity H,, expressed as a function of those elements.

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If we choose to introduce into the expression (T2.), for d H,, the variation of the time t, we have only to change de to da-ot, because, by (Q2.), d t enters only so accompanied ; that is, t enters only under the form t – Tis in the expressions of

Sini Lu x'; Y';'; as functions of the time and of the elements; we have, therefore,

8 H,
Σ.m μ' ; .

(211.) and since, by (H2.), (Q2.), H, =Σ.m μ,

(212.) we find finally,

8 H,

(U?.) dt

8t This remarkable form for the differential of HỊ, considered as a varying element, is general for all problems of dynamics. It may be deduced by the general method from the formulæ of the 13th and 14th numbers, which give ô H δ Η, δ και,

(8 Н, 8 x6 и 8 H, 8x6


δη δο

(213.) δ και,

8 H,

+ +
8 x, gt
ох, в

8x6m at X1 X2 .. Hon being any 6 n elements of a system expressed as functions of the time and of the quantities nd; or more concisely by this special consideration, that H, +H, is constant in the disturbed motion, and that in taking the first total differential coefficient of H, with respect to the time, the elements may by (F1.) be treated as constant. It is also a remarkable corollary of the general principles just referred to, but one not


of difficult to verify, that the first partial differential coefficient any


Et taken with respect to the time, may be expressed as a function of the elements alone, not involving the time explicitly.

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On the essential distinction between the Systems of Varying Elements considered in this

Essay and those hitherto employed by mathematicians. 37. When we shall have integrated the differential equations of varying elements (S2.), we can then calculate the varying relative coordinates gn , for any binary system (m, M), by the rules of undisturbed motion, as expressed by the equations (12.), (Q2.), or by the following connected formulæ :


r (cos o + I sin (8 — v) sin v), ,
(sin 0 – À sin (0 — ») cosv),

(V2.) &=2x – 0

; 22 sin (0 – v):

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in which the distance r is determined as a function of the time t and of the elements T, X, Mi, by the 5th equation (Q2.), and in which

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Ndr2 • g2 +

(W2) M + m x

() –-M q being still the minimum of r, when the orbit is treated as constant, and being still connected with the elements x, y, by the first equation of condition (186.). In astronomical language, M is the sun, m a planet, $n& are the heliocentric rectangular coordinates, r is the radius vector, 0 the longitude in the orbit, w the longitude of the perihelion, v of the node, 0 - w is the true anomaly, 0 - » the argument of latitude,

v u the constant part of the half square of undisturbed heliocentric velocity, diminished in the ratio of the sun's mass (M) to the sum (M + m) of masses of sun and planet, x is the double of the areal velocity diminished in the same ratio, is the versed sine of the inclination of the orbit, q the perihelion distance, and 7 the time of perihelion passage. The law of attraction or repulsion is here left undetermined ; for Newton's law, w is the sun's mass divided by the axis major of the orbit taken negatively, and

f x is the square root of the semiparameter, multiplied by the sun's mass, and divided by the square root of the sum of the masses of sun and planet. But the varying ellipse or other orbit, which the foregoing formulæ require, differs essentially (though little) from that hitherto employed by astronomers : because it gives correctly the heliocentric coordinates, but not the heliocentric components of velocity, without differentiating the elements in the calculation ; and therefore does not touch, but cuts,

; (though under a very small angle,) the actual heliocentric orbit, described under the influence of all the disturbing forces.

38. For it results from the foregoing theory, that if we differentiate the expressions (V2.) for the heliocentric coordinates, without differentiating the elements, and then assign to those new varying elements their values as functions of the time, obtained from the equations (S2.), and deduce the centrobaric components of velocity by the formulæ (12.), or by the following:


M + m?

y' =
M + m? M + mi

(214.) then these centrobaric components will be the same functions of the time and of the new varying elements which might be otherwise deduced by elimination from the integrals (Q2.), and will represent rigorously (by the extension given in the theory to those last-mentioned integrals) the components of velocity of the disturbed planet m, relatively to the centre of gravity of the whole solar system. We chose, as more suitable to the general course of our method, that these centrobaric components of velocity should be the auxiliary variables to be combined with the heliocentric coordinates, and to have their disturbed values rigorously expressed by the formulæ

x =

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of undisturbed motion ; but in making this choice it became necessary to modify these latter formulæ, and to determine a varying orbit essentially distinct in theory (though little differing in practice) from that conceived so beautifully by LAGRANGE. The orbit which he imagined was more simply connected with the heliocentric motion of a single planet, since it gave, for such heliocentric motion, the velocity as well as the position ; the orbit which we have chosen is perhaps more closely combined with the conception of a multiple system, moving about its common centre of gravity, and influenced in every part by the actions of all the rest. Whichever orbit shall be hereafter adopted by astronomers, they will remember that both are equally fit to represent the celestial appearances, if the numeric elements of either set be suitably determined by observation, and the elements of the other set of orbits be deduced from these by calculation. Meantime mathematicians will judge, whether in sacrificing a part of the simplicity of that geometrical conception on which the theories of LAGRANGE and Poisson are founded, a simplicity of another kind has not been introduced, which was wanting in those admirable theories ; by our having succeeded in expressing rigorously the differentials of all our own new varying elements through the coefficients of a single function : whereas it has seemed necessary hitherto to employ one function for the Earth disturbed by Venus, and another function for Venus disturbed by the Earth.

If we put,

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8 H, ov

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Integration of the Simplified Equations, which determine the new varying Elements.

39. The simplified differential equations of varying elements, (S2.), are of the same form as the equations (A.), and may be integrated in a similar manner. for abridgement,

8 H,
(t, x, v) = S* { $("H

+ x :)

(X2) and interpret similarly the symbols (fu, w, a), &c., we can easily assign the variations of the following 8 combinations, (, x, v) (th, w, 1) (yu, x, v) (7,w, 1) (, w, v) (fu, x, a)

%, (t, x, a) (fu, w, v); namely,

8 (1,4,v) = £. m (70 50 O Mo + xdw — xg wo toda – vodno) – H,dt,
δ (μ, ω,λ) = Σ.m (με δτο - μoτ + ωδ κι – ωδη +λαδνη -λδν) - Η, δε,
(, ,)= (

twoo — + ) - ,
ò (yu, x,y) = . m (po
= Σ. m (μο δτo - μoτ +κδω -- κυδωρ +νδλ – να δλ.) - Η, δε,

+ A – –
δ(τ,ω,λ) = Σ. m (τεμ το δμο + ω, δ κι –ωδη + λαδνη - λδν)
To + wg – ) – Hot,

- ,

(Y?.) δ (τ, ω, ν) = Σ. m (τεμ – το δμο + ω, δ - ωδη +νδλ - ν δ λο) – Η, δε,

δ (μ, κλ) Σ. m (με - μδτ
o (p,2,1)=&. m (od 7o MÓT + xow – no diao thodno - nòu) - Hedt,

- wo –
δ(τ, ,λ) = . m (τεμ κδω
ò (1,x,)= £ . m (Top To Mo + xow — xod we thodro 200) - Hydt,

wo + —
δ (μ, ω, ν) = Σ. m (με δτο - μδα + ως δκι - ωδη +νδλ – η δλ.) ) – H,ät,

Hat Yo ho Mo Yo To Wo being the initial values of the varying elements na me u gw. If, then, we consider, for example, the first of these 8 combinations (7, n, v), as a function of


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all the 3 n — 3 elements M; W; hi, and of their initial values wo, i Wo, i no, i, involving also

; in general the time explicitly, we shall have the following forms for the 6 n - 6 rigorous integrals of the 6 n-6 equations (S2.):

m; 7; = m. (t, x; v); m; 7o, i = -

(T, X,); 7


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و1- 1- 1-1


m; x; = 8w; (t, x, y); m;xo, i = -
(T,%, ');

M; Y; = 87; (5,4,1); m;Po, i = -

(T, %, v); and in like manner we can deduce forms for the same rigorous integrals, from any one of the eight combinations (Y2.). The determination of all the varying elements would therefore be fully accomplished, if we could find the complete expression for any one of these 8 combinations.

40. A first approximate expression for any one of them can be found from the form under which we have supposed H, to be put, namely, as a function of the elements and of the time, which may be thus denoted: H2 = H, (t, %, 21, Men, V1, 71, W1, ... Mn-1, An- 19 Min-1, 'n–1979-19 Wn-1); . 11

(A3.) by changing in this function the varying elements to their initial values, and employing the following approximate integrals of the equations (S2.),


ot 8 H,
ju = Ho + So


dt, T = TO
at 8 H,

ot 8 H,
w = w + S
= +
dt, x = x

(B3.) H

nt / H. a = no + Sodt

, v= 1 – = λ +

dt. For if we denote, for example, the first of the 8 combinations (Y2.) by G, so that G= {T, %, »}, : }

(C3.) we shall have, as a first approximate value,

H, δ Η, ]



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and after thus expressing G, as a function of the time, and of the initial elements, we can eliminate the initial quantities of the forms To Ho Yo, and introduce in their stead the final quantities we wn, so as to obtain an expression for G, of the kind supposed in (Z2.), namely, a function of the time t, the varying elements we w 2, and their initial values to wo ho. An approximate expression thus found may be corrected by a process of that kind, which has often been employed in this Essay for other similar purposes. For the function G, or the combination (T, , v), must satisfy rigorously, by (Y2.) (A3.), the following partial differential equation :

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