(674.) From the momentary changes in the elements of the disturbed orbit corresponding to successive situations of P and M, to conclude the total amount of change produced in any given time is the business of the integral calculus, and lies far beyond the scope of the present work. Without its aid, however, and by general considerations of the periodical recurrence of configurations of the same character, we have been able to demonstrate many of the most interesting conclusions to which geometers have been conducted, examples of which have already been given in the reasoning by which the permanence of the axes, the periodicity of the inclina tions, and the revolutions of the nodes of the planetary orbits have been demonstrated. We shall now proceed to apply similar considerations to the motion of the apsides, and the variations of the excentricities. To this end we must first trace the changes induced on the disturbing forces themselves, with the varying positions of the bodies, and here as in treating of the inclinations we shall suppose, unless the contrary is expressly indicated, both orbits to be very nearly circular, without which limitation the complication of the subject would become too embarrassing for the reader to follow, and defeat the end of explanation.

(675.) On this supposition the directions of S P and SY, the perpendicular on the tangent at P, may be regarded as coincident, and the normal and radial disturbing forces become nearly identical in quantity, also the tangential and transversal, by the near coincidence of the points T and L (fig. art. 687.).

So far then as the intensity of the forces is concerned, it will make very little difference in which way the forces are resolved, nor will it at all materially affect our conclusions as to the effects of the normal and tangential forces, if in estimating their quantitative values, we take advantage of the simplification introduced into their numerical expression by the neglect of the angle PSY, i. e. by the substitution for them of the radial and transversal components. The character of these effects depends (art. 670, 671.) on the direction in which the forces act, which we shall suppose normal and tangential as before, and it is only on the estimation of their quantitative effects that the error induced by the neglect of this angle can fall. In the lunar orbit this angle never exceeds 3° 10′, and its influence on the quantitative estimation of the acting forces may therefore be safely neglected in a first approximation. Now MN being found by the proportion M P2 : M S2 :: MS: MN, NP (=M N-MP) is also known, and therefore NL NP. sin NP S=N P. sin (A S P+SM P) and LS PL-PS=N P. cos NPS-PS-NP. cos (ASP + SMP)-SP become known, which express respectively the tangential and normal forces on the same scale that S M represents M's attraction on S. Suppose P to revolve in the direction E A D B. Then, by drawing the figure in various situations of P throughout the whole circle, the reader will easily satisfy himself -1st. That the tangential force accelerates P, as it moves from E towards A, and from D towards B, but retards it as it passes from A to D, and from B to E. 2nd. That the tangential force vanishes at the four points A, D, E, B, and attains a maximum at some intermediate points. 3rdly. That the normal force is directed outwards at the

=

· ( 1 +

1+

=

*

R

MSR; SP=r; M P=f; AS P=0; AM P= M; MN= ; NP

Ꭱ

(R

f

R R$ + + ; whence we have N L=(R-ƒ). sin (0+ M, f f

R Ꭱ

-T. When R

R+); LS=(R-J). cos (8+ M). (1++)-,

and f, owing to the great distance of M, are nearly equal, we have R-f= PV,1 nearly, and the angle M may be neglected; so that we have

R

NP-3 PV.

F F

syzigies A, B, and inwards at the points D, E, at which points respectively its outward and inward intensities attain their maxima. Lastly, that this force vanishes at points intermediate between A D, D B, B E, and E A, which points, when M is considerably remote, are situated nearer to the quadrature than the syzygies.

(676.) In the lunar theory, to which we shall now proceed to apply these principles, both the geometrical representation and the algebraic expression of the disturbing forces admit of great simplification. Owing to the great distance of the sun M, at whose center the radius of the moon's orbit never subtends an angle of more than about 8', N P may be regarded as parallel to A B. And D SE becomes a straight line coincident with the line of quadratures, so that V P becomes the cosine of ASP to radius SP, and NL=NP. sin ASP; LP NP. cos A S P. Moreover, in this case (see the note on the last article) N P 3 P V-3 SP. cos ASP; and consequently NL-3 SP.cos ASP. sin ASP = SP. sin 2 AS P, and L S=SP (3. cos A S P2 - 1) =SP (1+3. cos 2 ASP) which vanishes when cos AS P2, or at 64° 14′ from the syzygy. Suppose through every point of P's orbit there be drawn SQ=3 SP. cos A S P2, then will Q trace out a certain looped oval, as in the figure, cutting the orbit in four points 64° 14′ from A and B respectively, and PQ will always represent in quantity and direction the normal force acting at P.

(677.) It is important to remark here, because upon this the whole lunar theory and especially that of the motion of

the apsides hinges, that all the acting disturbing forces, at equal angles of elongation A SP of the moon from the sun, are cæteris paribus proportional to SP, the moon's distance from the earth, and are therefore greater when the moon is near its apogee than when near its perigee; the extreme proportion being that of about 28:25. This premised, let us first consider the effect of the normal force in displacing the lunar apsides. This we shall best be enabled to do by examining separately those cases in which the effects are most strongly contrasted; viz. when the major axis of the moon's orbit is directed towards the sun, and when at right angles to that direction. First, then, let the line of apsides be

directed to the sun as in the annexed figure, where A is the perigee, and take the arcs A a, Ab, Bc, B d each 64° 14'. Then while P is between a and b the normal force acting outwards, and the moon being near its perigee, by art. 671. the apsides will recede, but when between c and d, the force there acting outwards, but the moon being near its apogee, they will advance. The rapidity of these movements will be respectively at its maxima at A and B, not only because the disturbing forces are then most intense, but also because (see art. 671.) they act most advantageously at those points to displace the axis. Proceeding from A and B towards the neutral points a b c d the rapidity of their recess and advance diminishes, and is nothing (or the apsides are stationary) when P is at either of these points. From b to D, or rather to a point some little beyond D (art. 671.) the force acts inwards, and the moon is still near perigee, so that in this are

of the orbit the apsides advance. But the rate of advance is feeble, because in the early part of that arc the normal force is small, and as P approaches D and the force gains power, it acts disadvantageously to move the axis, its effect vanishing altogether when it arrives beyond D at the extremity of the perpendicular to the upper focus of the lunar ellipse. Thence up to c this feeble advance is reversed and converted into a recess, the force still acting inwards, but the moon now being near its apogee. And so also for the arcs d E, E a. In the figure these changes are indicated by for rapid advance, for rapid recess, + and — for feeble advance and recess, and 0 for the stationary points Now if the forces were equal on the sides of + and - it is evident that there would be an exact counterbalance of advance and recess on the average of a whole revolution. But this is not the case. The force in apogee is greater than that in perigee in the proportion of 28: 25, while in the quadratures about D and E they are equal. Therefore, while the feeble movements + and -in the neighbourhood of these points destroy each other almost exactly, there will necessarily remain a considerable balance in favour of advance, in this situation of the line of apsides.

(678.) Next, suppose the apogee to lie at A, and the perigee at B. In this case it is evident that, so far as the direction of the motions of the apsides is concerned, all the conclusions of the foregoing reasoning will be reversed by the substitution of the word perigee for apogee, and vice versâ ; and all the signs in the figure referred to will be changed. But now the most powerful forces act on the side of A, that is to say, still on the side of advance, this condition also being reversed. In either situation of the orbit, then, the apsides advance.

(679.) (Case 3.) Suppose, now, the major axis to have the situation D E, and the perigee to be on the side of D. Here, in the arc bc of P's motion the normal force acts inwards, and the moon is near perigee, consequently the apsides advance, but with a moderate rapidity, the maximum of the inward normal force being only half that of the outward.