X. On the Position of the Apsides of Orbits of great

Excentricity.

By W. WHEWELL, M.A. F.R.S.

FELLOW OF TRINITY COLLEGE, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Read April 17, 1820.]

of

THE problem of the determination of the path which a body will describe when acted upon by a force tending to any center is, as is well known, easily reduced to a differential equation. In order to learn the form and properties of the curves described, it would be necessary to integrate this, which integration can be effected only in particular cases. The cases for which this is possible, if the central force be supposed to vary as a simple power the distance, are the cases of nature, that is, when the force varies inversely as the square of the distance, the case when the force yaries inversely as the cube, and that when it varies directly as the distance. In these instances the orbit is known, and in two of them it may be a figure returning into itself, namely an ellipse. In the other cases we cannot accurately delineate the curve described, as we could if it were expressed by means of an algebraic equation : we know, however, in some degree, the form of the path. It will generally be a curve running round the center, and alternately

Besides these cases, we can integrate when the velocity is that which would be acquired in falling from infinity, whatever be the power according to which the force varies; and, under certain conditions, when the force varies inversely as the fifth power of the distance.

approaching to and receding from that point in loops perpetually similar; not returning into itself, but making its farthest excursions from the center, every time in a different direction. From this account it will be seen, that we should have a general idea of the curve, if we knew, in each particular instance, the proportion of the greatest and least distances from the center, and the angle contained between them.

When the velocity and direction are very nearly those which are required for the description of a circle, the angle between the greatest and least, or apsidal, distances, has been determined by Newton and other writers on Mechanics: but in other cases, where the orbits are more excentric, as it may be called, we have nothing but analogy and conjecture. These however can only enable us to judge very imperfectly of the form of the orbit. For instance, we know that in the cases of forces varying directly as the distance, and inversely as the square of the distance, the angle between the apsides is the same whatever be the excentricity. Are we therefore to conclude that the same will be true, or approximately so, in other instances? so far as can be conjectured, the probability is rather against this, since the principle of the approximation depends on supposing the excentricity small. To determine whether or not this is the case, and if not, to obtain limits within which the true value lies, is the object of the present paper. The question may be considered as of some importance, for it is manifest that if it became necessary actually to determine the orbit of a body under given circumstances, it would, in far the majority of cases, fall out of the bounds of the method of approximation above mentioned. The solution of the problem will also enable us to decide how far Newton was correct in the assumption that the angle to which we tend when we make the excentricity indefinitely small, may be taken for its value when the excentricity is small but finite. And if this

supposition be inaccurate, we may see in what direction the error lies, and form some idea of its magnitude.

It appears allowable to take for granted, that, knowing the angle between the apsides when the excentricity is indefinitely small, if we can also determine it when the excentricity is indefinitely great, it will for all intermediate excentricities have a value between these two. We shall proceed therefore to find it for the latter of these two cases.

Suppose a body to be projected in a direction perpendicular to its distance from a given center of force. If the velocity of projection be nearly equal to the velocity in a circle at the same distance, the body will revolve, continuing always nearly at the same distance from the centre. But if the velocity of projection be very small, it will (except when the force varies in a high inverse ratio) pass very near the center and then rise again to its former distance, having described an angle round the center which varies with the law of force, and, as appears probable, with the velocity of projection. We are now seeking this angle when the velocity of projection is indefinitely diminished.

First, suppose the force to vary directly as some power of the distance. Let a be the distance of projection, and at x, any other distance, let the force be equal to xn Also let 2 q to √(n+1) be

an

the ratio of the velocity of projection perpendicular to the radius to the velocity in a circle at the same distance. Then, if ✪ be any angle described about the center, and x the radius vector of the orbit, we shall easily find that

In order to find the nearest distance of the body from the center,

1)

(a"+1 — x" + 1) x2 — q2 a”+ 1 (a2 — x2) = 0........(2).

the roots of which equation are the apsidal distances.

One root of this equation is obviously a, and hence this is an apsidal distance, which agrees with the conditions of projection. Also, if n be odd, -a, is a root of the equation; but it is obvious that no negative quantity can answer our purpose. Now if we take the limiting equation of equation (2), we shall find that its possible roots are x=0; x = a positive quantity; and, in the case when n is odd, x = a negative quantity. Hence the roots of (2) are a, and, if n be odd, -a, and besides these a positive and a negative quantity, of which the first is the other apsidal distance.

Now when q=0, equation (2) becomes (a"+1 — x2 + 1) x2 x2+1) x2 = 0, which has two roots, x = 0, x = 0. Hence the two values of x in equation (2) which we are seeking, are those which arise from the two just mentioned, by introducing q. And when q is very small, it is clear that these will be very small; and we may thus approximate to them. By dividing by a"+12 the equation becomes

now when q and a are very small, the second and fourth terms of this are necessarily very small, and therefore the first and third must destroy each other approximately; that is

of these x = aq gives the lower apsidal distance.

Hence a factor of equation (2) is x2 only q1 &c.

a2 q2 + ẞ; where ß.involves

In fact, if we consider the first side of equation (2),

2

as composed of two factors [(1+g2) — a2 q2+d] and a+1-2 +1 + A q2 + B q*+Cq°+ &c. we shall find that A=- x2-1 (α2 — x2); B=- x2¬3 («2 — x2)2, &c. And will not involve any powers of q lower than qa+3 when n is odd, and q′′ x when n is even: and hence the value a q

is accurate as far as quantities of the order q*+2.