the theory of gravity. A good general idea of its nature may be formed by considering the direct action of the disturbing forces on the moon, supposed to move in a circular orbit. In such an orbit undisturbed, the velocity would be uniform; but the tangential force acting to accelerate her motion through the quadrants preceding her conjunction and opposition, and to retard it through the alternate quadrants, it is evident that the velocity will have two maxima and two minima, the former at the syzygies, the latter at the quadratures. Hence at the syzygies the velocity will exceed that which corresponds to a circular orbit, and at quadratures will fall short of it. The true orbit will therefore be less curved or more flattened than a circle in syzygies, and more curved (i. e. protuberant beyond a circle) in quadratures. This would be the case even were the normal force not to act. But the action of that force increases the effect in question, for at the syzygies, and as far as 64° 14' on either side of them, it acts outwards, or in counteraction of the earth's attraction, and thereby prevents the orbit from being so much curved as it otherwise would be; while at quadratures, and for 25° 46′ on either side of them, it acts inwards, aiding the earth's attraction, and rendering that portion of the orbit more curved than it otherwise would be. Thus the joint action of both forces distorts the orbit from a circle into a flattened or elliptic form, having the longer axis in quadratures, and the shorter in syzygies; and in this orbit the moon moves faster than with her mean velocity at syzygy (i. e. where she is nearest the earth) and slower at quadratures where farthest. Her angular motion about the earth is therefore for both reasons greater in the former than in the latter situation. Hence at syzygy her true longitude seen from the earth will be in the act of gaining on her mean, in quadratures of losing, and at some intermediate points (not very remote from the octants) will neither be gaining nor losing. But at these points, having been gaining or losing through the whole previous 90°, the amount of gain or loss will have attained its maximum. Consequently at the octants the true longitude will deviate most from the mean in excess and defect, and the

inequality in question is therefore nil at syzygies and quadratures, and attains its maxima in advance or retardation at the octants, which is agreeable to observation.

(706.) Let us, however, now see what account can be rendered of this inequality by the simultaneous variations of the axis and excentricity as above explained. The tangential force, as will be recollected, is nil at syzygies and quadratures, and a maximum at the octants, accelerative in the quadrants EA and D B, and retarding in AD and BE. In the two former then the axis is in process of lengthening; in the two latter, shortening. On the other hand the normal force vanishes at (a, b, d, e) 64° 14′ from the syzygies. It acts outwards over e A a, b B d, and inwards over a Db and dEe. In virtue of the tangential force, then, the point H moves towards P when P is in AD, BE, and from it when in DB, EA, the motion being nil when at A, B, D, E, and most rapid when at the octant D, at which points, therefore, (so far as this force is concerned,) the focus H would have its mean situation. And in virtue of the normal focus, the motion of H in the direction HQ will be at its maximum of rapidity towards Q at A, or B, from Q at D or

E, and nil at a, b, d, e. It will assist us in following out these indications to obtain a notion of the form of the curve

really described by H, if we trace separately the paths which H would pursue in virtue of either motion separately, since its true motion will necessarily result from the superposition of these partial motions, because at every instant they are at right angles to each other, and therefore cannot interfere. First, then, it is evident, from what we have said of the tangential force, that when P is at A, H is for an instant at rest, but that as P removes from A towards D, H continually approaches P along their line of junction HP, which is, therefore, at each instant a tangent to the path of H. When P is in the octant, H is at its mean distance from P (equal to PS), and is then in the act of approaching P most rapidly. From thence to the quadrature D the movement of H towards P decreases in rapidity till the quadrature is attained, when H rests for an instant, and then begins to reverse its motion, and travel from P at the same rate of progress as before towards it. Thus it is clear that, in virtue of the tangential force alone, H would describe a four-cusped curve a, d, b, e, its direction of motion round S in this curve being opposite to that of P, so that A and a, D and d, B and b, E and e, shall be corresponding points.

(707.) Next as regards the normal force.

When the

moon is at A the motion of H is towards D, and is at its

maximum of rapidity, but slackens as P proceeds towards D and as Q proceeds towards B. To the curve described, HIQ will be always a tangent, and since at the neutral point of the normal force (or when P is 64° 14′ from A, and Q 64° 14' from D), the motion of H is for an instant nil and is then reversed, the curve will have a cusp at 7 corresponding, and H will then begin to travel along the arc 1 m, while P describes the corresponding arc from neutral point to neutral point through D. Arrived at the neutral point between D and B, the motion of H along QH will be again arrested and reversed, giving rise to another cusp at m, and so on. Thus, in virtue of the normal force acting alone, the path of H would be the four-cusped, elongated curve l m n o, described with a motion round S the reverse of P's, and having a, d, b, e for points corresponding to A, B, D, E, places of P.

2

(708.) Nothing is now easier than to superpose these motions. Supposing H1, H2 to be the points in either curve corresponding to P, we have nothing to do but to set from off S, Sh equal and parallel to SH, in the one curve and from h, h H equal and parallel to SH, in the other. Let this be done for every corresponding point in the two curves, and there results an oval curve a d be, having for its semiaxes Sa=Sa, + Sa2; and Sd=Sd,+Sd2. And this will be the true path of the upper focus, the points a, d, b, e, corresponding to A, D, B, E, places of P. And from this it follows, 1st, that at A, B, the syzygies, the moon is in perigee in her momentary ellipse, the lower focus being nearer than the upper. 2dly, That in quadratures D, E, the moon is in apogee in her then momentary ellipse, the upper focus being then nearer

than the lower. 3dly, That H revolves in the oval a d be the contrary way to P in its orbit, making a complete revolution from syzygy to syzygy in one synodic revolution of the moon.

(709.) Taking 1 for the moon's mean distance from the earth, suppose we represent Sa, or Sd, (for they are equal) by 2a, Sa, by 2b, and Sd, by 2c, then will the semiaxes of the oval ad be, Sa and Sd be respectively 2a+2b and 2a+2c, so that the excentricities of the momentary ellipses at A and D will be respectively a+b and a+c. The total amount of the effect of the tangential force on the aris, in passing from syzygy to quadrature, will evidently be equal to the length of the curvilinear arc a, d, (fig. art. 708.), which is necessarily less than Sa1 + Sd, or 4a. Therefore the total effect on the semiaris or distance of the moon is less than 2a, and the excess and defect of the greatest and least values of this distance thus varied above and below the mean value S A=1 (which call a) will be less than a. The moon then is moving at A in the perigee of an ellipse whose semiaxis is 1+a and excentricity a+b, so that its actual distance from the earth there is 1+a-a-b, which (because a is less than a) is less than 1-b. Again, at D it is moving in apogee of an ellipse whose semiaxis is 1-a and excentricity a+c, so that its distance then from the earth is 1-a+a+c, which (a being greater than a) is greater than 1+c, the latter distance exceeding the former by 2a−2a+b+c.

(710.) Let us next consider the corresponding changes induced upon the angular velocity. Now it is a law of elliptic motion that at different points of differentellipses, each differing very little from a circle, the angular velocities are to each other as the square roots of the semiaxes directly, and as the squares of the distances inversely. In this case the semiaxes at A and D are to each other as 1+ a to 1-a, or as 1 : 1 - 2a, so that their square roots are to each other as 1 : 1 -a. Again, the distances being to each other as 1+a-a-b: 1— a+a+c, the inverse ratio of their squares (since a, a, b, c, are all very small quantities) is that of 1-2a +2a + 2c: 1+ 2a-2a-2b, or as 1:1-4a-4a-2b-2c. The angular